Quadratic Formula Calculator with Steps (Free) | StatsUnlock

A quadratic formula calculator solves ax² + bx + c = 0 by putting a, b and c into one formula.

Example: for 3x² − 5x − 2 = 0, a = 3, b = −5 and c = −2. The discriminant is 25 + 24 = 49, so x = (5 ± 7) ÷ 6. The roots are x = 2 and x = −13.

Rule of thumb: look at D = b² − 4ac first. If D is positive there are 2 real roots, if D = 0 there is 1 repeated root, and if D is negative there are no real roots, only 2 complex ones.

A quadratic formula calculator is a tool that finds the roots of a quadratic equation. A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a is not 0. The roots (also called solutions or zeros) are the values of x that make the equation true.

The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a. You read off three numbers, a, b and c, and the formula returns the roots. It never fails, which is why it is the method most students trust in an exam.

A worked one-liner: in 3x² − 5x − 2 = 0 we have a = 3, b = −5, c = −2. Then b² − 4ac = 25 + 24 = 49, √49 = 7, and x = (5 ± 7) ÷ 6. So x = 2 or x = −1/3.

What this calculator reports:

• Exact roots, as whole numbers, fractions, simplified square roots or complex numbers a ± bi.
• Decimal roots to 2, 4, 6 or 10 places.
• The discriminant and the nature of the roots.
• Every step, generated by the same code that computes the answer.
• A check: each root is substituted back into the equation.
• The graph facts: vertex, axis of symmetry, y-intercept, factored form and vertex form.

Who uses it: students in Algebra 1, Algebra 2, GCSE, IGCSE, CBSE Class 10, SAT and ACT courses; teachers building answer keys; and engineers, physicists and programmers who need a quick exact root for a projectile, an area or a break-even problem.

Formula, factoring or completing the square? All three solve the same equation. They differ in when they are fastest.

The formula itself comes from completing the square on ax² + bx + c = 0 once, with letters. Methods for solving quadratics go back to Babylonian tablets and to al-Khwarizmi's algebra in the 9th century (Gandz, 1937; Katz, 2007).

Nothing is calculated until you press Solve. Sample 1 is loaded so you can see the format.

4.1 What the answer means

___ Solve a problem above to fill this in with your own numbers.

A root is a number you can put in place of x so that the left side of the equation becomes exactly 0. A quadratic has at most two of them. If you draw y = ax² + bx + c, the real roots are the x values where the curve touches or crosses the horizontal axis.

4.2 What the discriminant tells you

___ Solve a problem above.

The discriminant is D = b² − 4ac, the part of the quadratic formula under the square root. Its sign decides the type of answer, so work it out first.

4.3 Exact answer or decimal answer

___ Solve a problem above.

An exact answer such as (3 + √13) ÷ 4 is the true root. A decimal such as 1.6514 is a rounded copy of it. Use the exact form in algebra, proofs and "leave your answer in surd form" questions. Use the decimal form for lengths, times, money and anything you will measure or plot.

4.4 What the graph shows

___ Solve a problem above.

The vertex is at x = −b ÷ 2a, the first half of the quadratic formula. The second half, √D ÷ 2a, is how far each root sits to the left and right of that line. That is why the two roots are always mirror images about the axis of symmetry.

4.5 How the answer changes when c changes

___ Solve a problem above.

Changing c slides the whole parabola up or down without changing its shape. Slide an upward parabola high enough and it lifts off the axis: the two roots move together, meet, and then stop being real.

4.6 How to check the answer yourself

There are two quick checks. First, substitute each root back into the original equation and confirm you get 0. Second, use the sum and product rules: the roots must add to −b ÷ a and multiply to c ÷ a. For 3x² − 5x − 2 = 0 the roots 2 and −1/3 add to 5/3 and multiply to −2/3, which matches −(−5) ÷ 3 and −2 ÷ 3.

4.7 Special cases

• a = 0: the equation is linear, not quadratic. The formula would divide by 0. Solve bx + c = 0 instead.
• b = 0: ax² + c = 0 gives x = ±√(−c ÷ a). The roots are equal and opposite.
• c = 0: one root is always 0. Factor out x: x(ax + b) = 0. Never divide both sides by x, or you lose that root.
• Very large or very small coefficients: on a computer the plain formula can lose digits when b² is much bigger than 4ac. This calculator uses exact integer arithmetic, so it does not have that problem (Goldberg, 1991; Forsythe, 1970).

4.8 Where this leads next

The same discriminant idea returns in conic sections, in the roots of cubic equations, and in physics whenever a quantity depends on time squared. If you are comfortable reading D, you can tell how many times a ball reaches a given height or whether a line meets a circle before you calculate anything.

▶ Solve a problem above and all five cards fill in with your own answer.

This flowchart shows how the quadratic formula is applied step by step, from the raw equation to checked roots. Every box carries the worked number for 3x² − 5x − 2 = 0, the same example used in the Excel, R, Python and by-hand sections.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return x = 2 and x = −0.3333. Section 14 does the same problem by hand and on a Casio or TI calculator.

1. Choose how to enter the problem. Pick a tab: Type the Equation, Enter a, b, c, From 3 Points, or Batch (CSV).
2. Type your quadratic equation. Write it as you would on paper, for example 3x^2 - 5x - 2 = 0. You do not need to rearrange it first.
3. Or load a sample problem. Open the Sample problem list and choose one of the ten named examples to see the format.
4. Set the decimal places. Choose 2, 4, 6 or 10 decimal places for the rounded roots.
5. Choose the answer form and number system. Keep Exact and decimal for surds and fractions, and switch to Real numbers only if your course does not use complex numbers.
6. Press Solve. Nothing is calculated until you press Solve or press Enter in an input box.
7. Read the answer box. The exact roots come first, the decimals under them, with a badge for the type of roots and a Verified badge.
8. Follow the step-by-step solution. Each step shows one move: standard form, the discriminant, the quadratic formula, the two roots and the check.
9. Study the table and the four charts. See the vertex, axis of symmetry, factored form and vertex form, and look at the graph with the roots marked.
10. Download or copy the working. Use Download Doc for a text file with every step, Download PDF to print, or the Copy buttons in Section 5.

Excel has no single quadratic function, but the formula fits in two short cells. Put a, b and c in one row, work out the discriminant, then write one formula for each root. The steps below use the same example as the rest of the page, 3x² − 5x − 2 = 0. They work the same way in Google Sheets.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

• Negative discriminant. SQRT of a negative number returns #NUM!. Use =IMDIV(IMSUM(-B2,IMSQRT(D2)),2*A2), which returns text such as −1+2i. For the second root use IMSUB in place of IMSUM.
• a = 0. The formula divides by 2*A2 and returns #DIV/0!. That is correct: the equation is linear.
• Huge b compared with a and c. Excel keeps about 15 significant digits. For x² − 1000000x + 1 = 0 the small root comes out as 1.00000761E-06 where the true value is 1.000000000001E-06, so the sixth digit is already wrong. Use the stable form: put =-(B2+SIGN(B2)*SQRT(D2))/2 in a helper cell q, then the roots are q/A2 and C2/q.
• Many equations. Put one equation per row and fill the formulas down. That is a ready-made answer key.
• Goal Seek. Data, What-If Analysis, Goal Seek can also find a root, but it returns only the one nearest your starting guess, and only as a decimal.

9.4 Why Excel might disagree with R or Python

9.5 Charting it in Excel

Make a column of x values from about one unit left of the smaller root to one unit right of the larger root, in steps of 0.1. Next to it compute =$A$2*x^2+$B$2*x+$C$2. Select both columns and insert a Scatter with Smooth Lines chart. Do not use a Line chart: it spaces the x values evenly as labels and distorts the parabola. Then set the horizontal axis to cross at 0 so the roots are easy to read.

9.6 Excel errors you will probably hit

R can solve a quadratic in one line with polyroot(), but writing the formula out shows the discriminant and the nature of the roots as well. The script below does both, checks the answer and draws the graph. It uses base R only, so there is nothing to install.

10.1 The complete script

R, base only, saves quadratic_formula_plot.png

# ============================================================
# Quadratic formula calculator in R (base R only)
# Solves a*x^2 + b*x + c = 0
# Worked example: 3x^2 - 5x - 2 = 0  ->  x = -1/3 or x = 2
# ============================================================

# ---- 1. The problem: change only these three numbers ----
a <- 3
b <- -5
c <- -2

if (a == 0) stop("a must not be 0. With a = 0 the equation is linear.")

# ---- 2. Discriminant and nature of the roots ----
D <- b^2 - 4 * a * c
nature <- if (D > 0) {
  "two different real roots"
} else if (D == 0) {
  "one repeated real root"
} else {
  "two complex conjugate roots (no real roots)"
}

# ---- 3. Roots with the quadratic formula ----
# as.complex() lets sqrt() return an imaginary number when D is negative
sqrtD <- sqrt(as.complex(D))
x1 <- (-b - sqrtD) / (2 * a)
x2 <- (-b + sqrtD) / (2 * a)

# ---- 4. Numerically stable version (avoids cancellation) ----
q  <- -(b + ifelse(b >= 0, 1, -1) * sqrtD) / 2
s1 <- q / a
s2 <- if (q != 0) c / q else s1

# ---- 5. The same roots from polyroot(): ASCENDING order c, b, a ----
pr <- polyroot(c(c, b, a))

# ---- 6. Other results ----
h <- -b / (2 * a)            # axis of symmetry
k <- c - b^2 / (4 * a)       # vertex height

# ---- 7. Check by substituting back ----
f <- function(x) a * x^2 + b * x + c
residuals <- Mod(c(f(x1), f(x2)))

show <- function(z) if (abs(Im(z)) < 1e-12) format(round(Re(z), 4), nsmall = 4) else format(round(z, 4), nsmall = 4)

cat("Equation      :", a, "x^2 +", b, "x +", c, "= 0\n")
cat("Discriminant  : D =", D, "\n")
cat("Nature        :", nature, "\n")
cat("Roots         : x1 =", show(x1), ", x2 =", show(x2), "\n")
cat("Stable formula:", show(s1), ",", show(s2), "\n")
cat("polyroot()    :", show(pr[1]), ",", show(pr[2]), "\n")
cat("Sum, product  :", show(x1 + x2), "(= -b/a),", show(x1 * x2), "(= c/a)\n")
cat("Vertex        : (", round(h, 4), ",", round(k, 4), ")\n")
cat("Check         : residuals", signif(residuals[1], 3), "and", signif(residuals[2], 3), "\n")
verdict <- if (all(residuals < 1e-9)) "Verified" else "CHECK FAILED"
cat("Verdict       :", verdict, "- result:", nature, "\n")

# ---- 8. One 300 dpi figure: parabola, roots, vertex, axis ----
half <- max(abs(Re(x2) - Re(x1)) * 0.9, 1.5)
xs <- seq(h - half, h + half, length.out = 400)
png("quadratic_formula_plot.png", width = 2100, height = 1350, res = 300)
plot(xs, f(xs), type = "l", lwd = 2.5, col = "#2563eb",
     xlab = "x", ylab = "y", main = "Quadratic formula: roots, vertex and axis")
abline(h = 0, col = "#475569")
abline(v = h, col = "#9333ea", lty = 2)
points(h, k, pch = 18, cex = 1.8, col = "#9333ea")
if (D >= 0) points(c(Re(x1), Re(x2)), c(0, 0), pch = 19, cex = 1.5, col = "#16a34a")
grid(col = "#e5e7eb")
dev.off()
cat("Saved quadratic_formula_plot.png\n")

Expected output for the worked example: D = 49, roots x1 = −0.3333 and x2 = 2.0000, sum 1.6667, product −0.6667, vertex (0.8333, −4.0833), verdict Verified.

10.2 What each part does

10.3 What the figure shows

A blue parabola for y = 3x² − 5x − 2, a grey line at y = 0, green dots at the roots x = −1/3 and x = 2, a purple diamond at the vertex (0.8333, −4.0833), and a dashed purple axis of symmetry at x = 0.8333.

10.4 Common R problems and fixes

10.5 Useful one-liners

Python gives you both kinds of answer. SymPy returns the exact roots, such as −1/3 or 2 − √3, and NumPy returns fast decimals. The script below prints both, checks them and saves the graph.

Install with pip install sympy numpy matplotlib if you do not already have them.

11.1 The complete script

Python 3 with SymPy, saves quadratic_formula_plot.png

# ============================================================
# Quadratic formula calculator in Python
# Solves a*x^2 + b*x + c = 0 exactly (SymPy) and in decimals
# Worked example: 3x^2 - 5x - 2 = 0  ->  x = -1/3 or x = 2
# ============================================================
import cmath

import matplotlib
matplotlib.use("Agg")            # remove this line in Jupyter or Colab
import matplotlib.pyplot as plt
import numpy as np
import sympy as sp

# ---- 1. The problem: change only these three numbers ----
a, b, c = sp.Rational(3), sp.Rational(-5), sp.Rational(-2)

if a == 0:
    raise ValueError("a must not be 0. With a = 0 the equation is linear.")

# ---- 2. Discriminant and nature of the roots ----
D = b**2 - 4*a*c                 # ** is power in Python. ^ is XOR, not power.
if D > 0:
    nature = "two different real roots"
elif D == 0:
    nature = "one repeated real root"
else:
    nature = "two complex conjugate roots (no real roots)"

# ---- 3. Exact roots with the quadratic formula ----
x1 = sp.simplify((-b - sp.sqrt(D)) / (2*a))
x2 = sp.simplify((-b + sp.sqrt(D)) / (2*a))

# ---- 4. Decimal roots, numerically stable (no cancellation) ----
af, bf, cf = float(a), float(b), float(c)
sqrtD = cmath.sqrt(bf*bf - 4*af*cf)
q = -(bf + (sqrtD if bf >= 0 else -sqrtD)) / 2
r1 = q / af
r2 = cf / q if q != 0 else r1

# ---- 5. Other results ----
h = -b / (2*a)                   # axis of symmetry
k = c - b**2 / (4*a)             # vertex height
x = sp.symbols("x")
factored = sp.factor(a*x**2 + b*x + c)

def show(z):
    """Real numbers print as 2.0000, complex ones as -1.0000+2.0000j."""
    return f"{z.real:.4f}" if abs(z.imag) < 1e-12 else f"{z:.4f}"

# ---- 6. Check by substituting back ----
res1 = sp.simplify(a*x1**2 + b*x1 + c)
res2 = sp.simplify(a*x2**2 + b*x2 + c)

print(f"Equation      : {sp.expand(a*x**2 + b*x + c)} = 0")
print(f"Discriminant  : D = {D}")
print(f"Nature        : {nature}")
print(f"Exact roots   : x1 = {x1}, x2 = {x2}")
print(f"Decimal roots : x1 = {show(complex(sp.N(x1)))}, x2 = {show(complex(sp.N(x2)))}")
lo, hi = sorted([r1, r2], key=lambda z: (z.real, z.imag))
print(f"Stable formula: {show(lo)}, {show(hi)}")
print(f"numpy.roots   : {np.roots([af, bf, cf])}")   # descending order: a, b, c
print(f"Sum, product  : {x1 + x2} (= -b/a), {sp.simplify(x1*x2)} (= c/a)")
print(f"Vertex        : ({h}, {k})")
print(f"Factored form : {factored}")
print(f"Check         : residuals {res1} and {res2}")
verdict = "Verified" if res1 == 0 and res2 == 0 else "CHECK FAILED"
print(f"Verdict       : {verdict}. Result: {nature}.")

# ---- 7. One 300 dpi figure: parabola, roots, vertex, axis ----
hf, kf = float(h), float(k)
half = max(abs(float(sp.re(sp.N(x2))) - float(sp.re(sp.N(x1)))) * 0.9, 1.5)
xs = np.linspace(hf - half, hf + half, 400)
ys = af*xs**2 + bf*xs + cf

fig, ax = plt.subplots(figsize=(7, 4.5))
ax.plot(xs, ys, color="#2563eb", lw=2.5, label="y = ax² + bx + c")
ax.axhline(0, color="#475569", lw=1)
ax.axvline(hf, color="#9333ea", lw=1.2, ls="--", label="axis of symmetry")
ax.plot([hf], [kf], "D", color="#9333ea", ms=8, label=f"vertex ({h}, {k})")
if D >= 0:
    ax.plot([float(x1), float(x2)], [0, 0], "o", color="#16a34a", ms=9, label="roots")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_title("Quadratic formula: roots, vertex and axis of symmetry")
ax.grid(color="#e5e7eb")
ax.legend(frameon=False, fontsize=9)
fig.savefig("quadratic_formula_plot.png", dpi=300, bbox_inches="tight")
print("Saved quadratic_formula_plot.png")

Expected output for the worked example: D = 49, exact roots x1 = −1/3 and x2 = 2, decimal roots −0.3333 and 2.0000, sum 5/3, product −2/3, vertex (5/6, −49/12), factored form (x − 2)(3x + 1), verdict Verified. This script was run before publishing and printed exactly these values.

11.2 What each part does

11.3 What the figure shows

The same picture as the R script: the blue parabola, green dots at x = −1/3 and x = 2, a purple diamond at the vertex (5/6, −49/12) and the dashed axis of symmetry. When D is negative no green dots are drawn, because the curve never meets the axis.

11.4 Common Python problems and fixes

11.5 Useful one-liners

12.1 Discriminant cases

How to use it: compute D, find its row, and you know what kind of answer to expect before you finish.

12.2 Perfect squares from 1 to 30

How to use it: if your discriminant is in the n² column, the roots are rational and the quadratic factors.

12.3 Simplifying common square roots

How to use it: take the largest square factor out. √72 = √(36 × 2) = 6√2.

12.4 Sign patterns of a, b and c

12.5 Three forms of a quadratic

12.6 Powers of i for complex roots

Eight worked problems, each solved with the quadratic formula. They cover the cases you meet in class: clean roots, fractions, decimals, a real-world problem, a repeated root, complex roots, large numbers and a common trap.

This is the pen and paper method a teacher expects to see in an exam, worked on the same example as Sections 7, 9, 10 and 11: 3x² − 5x − 2 = 0. After it come the keystrokes for putting the quadratic formula into a Casio or TI calculator.

14.1 The written method

1. Write the equation in standard form, ax² + bx + c = 0. Move every term to one side. Here it is already done.
2. Write down a, b and c with their signs. a = 3, b = −5, c = −2.
3. Write the formula in letters. x = (−b ± √(b² − 4ac)) ÷ 2a. This line often earns a mark by itself.
4. Work out the discriminant first. D = (−5)² − 4(3)(−2) = 25 + 24 = 49. Put b in brackets before you square it.
5. Read the discriminant. D is positive and 49 = 7², so expect two rational roots.
6. Substitute. x = (−(−5) ± √49) ÷ (2 × 3) = (5 ± 7) ÷ 6.
7. Split the ±. x = (5 + 7) ÷ 6 = 12 ÷ 6 = 2, and x = (5 − 7) ÷ 6 = −2 ÷ 6 = −1/3.
8. Check. 3(2)² − 5(2) − 2 = 12 − 10 − 2 = 0. 3(−1/3)² − 5(−1/3) − 2 = 1/3 + 5/3 − 2 = 0.
9. Write the answer the way the question asks. x = 2 or x = −1/3. To 4 decimal places, x = 2.0000 or x = −0.3333.

14.2 How to put the quadratic formula into a calculator

There are three ways: type the formula directly, use the built-in equation mode, or store a short program. All three are shown for 3x² − 5x − 2 = 0.

Basic models such as the Casio fx-82MS and fx-82ES have no equation mode, so use the first row and type the formula. Menu numbers can differ between firmware versions and regional models. If your screen shows something different, check the equation chapter of the manual for your exact model.

A quadratic formula program for the TI-84. Press prgm, choose NEW, name it QUAD, and type these lines. sto→ gives the arrow, and Prompt and Disp are in the prgm I/O menu.

:Prompt A,B,C
:B²-4AC→D
:Disp "D=",D
:If D≥0
:Then
:Disp (-B+√(D))/(2A)
:Disp (-B-√(D))/(2A)
:Else
:Disp "NO REAL ROOTS"
:End

Run it with prgm, QUAD, enter. Type 3, −5 and −2 at the prompts and it shows D = 49, then 2 and −0.3333. Use the (-) key for negative numbers, not the subtraction key. Many exams allow stored programs and some do not, so check your exam rules first.

14.3 Practice problems with answers

Ten problems, easy to hard. Work each one on paper, then open the card to check. Every answer is exact.

14.4 Mistakes that lose marks

• The sign of −b. With b = −5, −b is +5, not −5.
• Squaring a negative b. (−5)² is 25. Typing −5² on a calculator gives −25.
• The sign of −4ac. When a and c have opposite signs, −4ac is positive and you add.
• Dividing only part of the top. The whole of −b ± √D is divided by 2a.
• Not rearranging first. 2x² = 4x + 3 has c = −3, not +3, once it is written as 2x² − 4x − 3 = 0.
• Rounding √D early. Keep the surd to the last line, then round once.
• Dividing both sides by x. This throws away the root x = 0.
• Stopping at one root. Unless D = 0, there are two.

The quadratic formula is the one method that always works. That does not make it the fastest every time.

Use the quadratic formula when:

• ✓ The discriminant is not a perfect square, so the roots are surds.
• ✓ The coefficients are decimals, fractions or large numbers.
• ✓ You tried to factor for a minute and found nothing.
• ✓ The question says "give your answer to 2 decimal places" or "in surd form".
• ✓ You expect complex roots.

Use something else when:

• ✗ The quadratic factors on sight, such as x² − 5x + 6. Factoring takes ten seconds.
• ✗ There is no x term, such as 3x² = 12. Divide and take a square root.
• ✗ There is no constant, such as x² − 4x = 0. Factor out x.
• ✗ You need the vertex or the minimum value. Complete the square.
• ✗ The equation is not a quadratic at all, for example it has x³ or 1/x.

Four real situations

A quick decision path

1. Is there no x term? Isolate x² and take ± the square root.
2. Is there no constant? Factor out x.
3. Compute D = b² − 4ac. Is it a perfect square? Then factor, or use the formula if you prefer.
4. Is D positive but not a square, or negative? Use the quadratic formula.
5. Do you need the turning point as well? Complete the square, or use h = −b ÷ 2a.

Assumptions

1. The equation is a quadratic in one variable. After moving everything to one side, the highest power must be 2. Check by expanding any brackets. If the x² terms cancel, the tool tells you the equation is linear.
2. The coefficients are real numbers you type exactly. A decimal such as 0.333 is read as 333/1000, not as 1/3. If you mean one third, type 1/3.
3. One letter stands for the unknown. x, t, n or any single letter works. Two different letters in one equation are rejected.
4. Roots are reported over the complex numbers by default. If your course uses real numbers only, switch the Number system box to "Real numbers only" and a negative discriminant is reported as "No real solution".
5. Decimals are rounded half away from zero to the number of places you choose. The exact form is never rounded.

Limitations

1. Degree 2 only. Cubic and higher equations need other methods, such as the rational root theorem or a numerical root finder.
2. No square roots or pi inside the coefficients. Type them as decimals. The answer is then exact for the decimal you typed, not for the surd you had in mind.
3. No division by an expression in x. For an equation such as 1/x + x = 3, multiply through by x yourself first, and remember that x = 0 is then excluded.
4. Very large square-free parts. Simplifying √D needs the square factors of D. For discriminants with more than about 13 digits the tool may leave a large surd partly unsimplified. The decimal roots stay correct.
5. Word problems still need you. The tool solves the equation. Deciding which root makes sense, for example rejecting a negative length, is your job.
6. Charts use ordinary floating-point numbers. For extreme coefficients the graph is a sketch. The printed roots are computed with exact integers and are not affected.

A quadratic formula calculator does one job well: it turns a, b and c into the roots of ax² + bx + c = 0, every time, whether or not the equation factors. The formula x = (−b ± √(b² − 4ac)) ÷ 2a is short enough to memorise and reliable enough to trust under exam pressure.

What the answer is

___ Solve a problem above to see your answer summarised here.

What it tells you

___ Solve a problem above.

The most useful habit is to compute the discriminant before anything else. Its sign tells you whether to expect two real roots, one repeated root or a complex pair, and whether it is a perfect square tells you if factoring would have worked. Students who do this catch most errors early, because a surprising discriminant is a warning that a sign went wrong.

Watch out for

___ Solve a problem above.

Three slips cause most wrong answers: forgetting that −b changes the sign of b, squaring a negative b without brackets, and dividing only part of the numerator by 2a. Research on how students solve quadratics finds the same pattern again and again (Vaiyavutjamai & Clements, 2006; Didiş & Erbaş, 2015). A ten-second check with the sum and product of the roots catches all three.

Next steps

___ Solve a problem above.

Type your own equation into the calculator above, read the steps once, and then cover them and repeat the working on paper. When your paper answer matches the verified answer three times in a row, you no longer need the quadratic formula calculator for that type of problem.

These peer-reviewed journal articles support the methods and the teaching advice in this quadratic formula calculator. They cover the history of the method, accurate computation of quadratic roots, and research on how students learn and mis-learn quadratic equations.

1. Aberth, O. (1973). Iteration methods for finding all zeros of a polynomial simultaneously. Mathematics of Computation, 27(122), 339–344. https://doi.org/10.1090/S0025-5718-1973-0329236-7
2. Blinn, J. F. (2005). How to solve a quadratic equation. IEEE Computer Graphics and Applications, 25(6), 76–79. https://doi.org/10.1109/MCG.2005.134
3. Blinn, J. F. (2006). How to solve a quadratic equation. Part 2. IEEE Computer Graphics and Applications, 26(2), 82–87. https://doi.org/10.1109/MCG.2006.35
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