Quadratic Formula Calculator

Quadratic equation solver

Find both roots, step by step

Put the equation in standard form, enter a, b, and c, then see the roots before the discriminant and substitution work.

Quadratic formula

x =−b ± √(b² − 4ac)2a

Your equationx² − 3x + 2 = 0

a: x² coefficient
b: x coefficient
c: constant

Enter coefficients

Match a, b, and c to ax² + bx + c = 0

Try an example

Result

Two solutions

First rootx₁ = 1Second rootx₂ = 2

D > 0

Two distinct real roots

The parabola crosses the x-axis twice.

DiscriminantD = 1

D < 0complexD = 0repeatedD > 0two real

Substitution

Follow the numbers through the formula

1
Write standard formx² − 3x + 2 = 0
a = 1, b = -3, c = 2
2
Calculate the discriminantD = (-3)² − 4(1)(2) = 1
D > 0, so two distinct real roots.
3
Substitute and evaluate both branchesx = (−(-3) ± √(1)) ÷ 2(1)
x₁ = 1 · x₂ = 2

Calculated in your browser with the tested quadratic formula.

Discriminant-first root analysis

Let the discriminant forecast a quadratic before evaluating its roots

The quadratic formula is universal for ax²+bx+c=0 when a is nonzero, but its real explanatory power comes from separating classification, substitution, and verification.

Step-by-step guide · Discriminant-first root analysis

A projectile-height model exposes both useful and extraneous roots

Standard form: ax² + bx + c = 0, a ≠ 0
Discriminant: D = b² − 4ac
Roots: x = (−b ± √D)/(2a)

Classify with D
D=20²−4(−5)(1.5)=400+30=430. Because 430 is positive, two distinct real algebraic roots will appear.
Keep the denominator grouped
x=(−20±√430)/(−10). Parentheses around 2a prevent the classic error of dividing only the square-root term by −10.
Interpret the modeled domain
The roots are approximately −0.074 and 4.074. Both solve the polynomial, but negative time lies before the modeled launch. The physically relevant ground time is therefore about 4.074 seconds.

Let the discriminant forecast a quadratic before evaluating its roots: Suppose height in meters is h(t)=−5t²+20t+1.5. To find when the object reaches ground level, solve −5t²+20t+1.5=0 with a=−5, b=20, and c=1.5.

Standard form makes every quadratic coefficient unambiguous

Before entering a, b, and c, move every term to one side, expand necessary products, and combine like powers. The missing x term in 3x²−12=0 means b=0, not b=−12; the missing constant in 2x²+5x=0 means c=0. If a=0, the equation is linear and belongs in the solve equation calculator rather than a quadratic formula form.

The discriminant D=b²−4ac is evaluated before its square root. D>0 produces two distinct real roots, D=0 produces one repeated real root, and D<0 produces a complex-conjugate pair. This classification is not a decorative label: it predicts the number of x-axis crossings of the corresponding parabola and tells you whether real-number arithmetic alone can finish the calculation.

A projectile-height model exposes both useful and extraneous roots

Suppose height in meters is h(t)=−5t²+20t+1.5. To find when the object reaches ground level, solve −5t²+20t+1.5=0 with a=−5, b=20, and c=1.5.

Classify with D
D=20²−4(−5)(1.5)=400+30=430. Because 430 is positive, two distinct real algebraic roots will appear.
Keep the denominator grouped
x=(−20±√430)/(−10). Parentheses around 2a prevent the classic error of dividing only the square-root term by −10.
Interpret the modeled domain
The roots are approximately −0.074 and 4.074. Both solve the polynomial, but negative time lies before the modeled launch. The physically relevant ground time is therefore about 4.074 seconds.

Equivalent quadratic methods reveal different structure

Factoring is fastest when structure is visible

For x²−5x+6=0, (x−2)(x−3)=0 exposes roots immediately. The broader algebra calculator can show this route for supported polynomial input.

Completing the square explains the formula

Rearranging a general quadratic into a squared binomial derives the ± branch and connects roots with vertex form. It is longer numerically but conceptually rich.

The formula handles every valid coefficient set

It does not depend on spotting integer factors. When D is negative, √D introduces i; the exponent calculator’s inverse-power mode likewise follows a declared real principal branch and is not a complex-root substitute.

Root verification catches substitution and sign errors

Rebuild standard form from the original equation and confirm that every coefficient includes its sign.
Check D independently; confusing −b² with (−b)² changes classification.
Substitute each displayed root into ax²+bx+c and confirm that the residual is zero or acceptably close after rounding.
Keep algebraic validity separate from contextual validity: lengths, dates, counts, and elapsed times may exclude a mathematically correct negative root.

Important limitations

Quadratic Formula Calculator is limited to the finite inputs, modes, and outputs stated for the Quadratic equation solver method. Check coefficient a, coefficient b, and coefficient c, validation messages, the equation, and the stated display precision before relying on discriminant, root type, first root, and second root.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Discriminant-first root analysis

Let the discriminant forecast a quadratic before evaluating its roots

The quadratic formula is universal for ax²+bx+c=0 when a is nonzero, but its real explanatory power comes from separating classification, substitution, and verification.

Step-by-step guide · Discriminant-first root analysis

A projectile-height model exposes both useful and extraneous roots

Standard form: ax² + bx + c = 0, a ≠ 0
Discriminant: D = b² − 4ac
Roots: x = (−b ± √D)/(2a)

Classify with D
D=20²−4(−5)(1.5)=400+30=430. Because 430 is positive, two distinct real algebraic roots will appear.
Keep the denominator grouped
x=(−20±√430)/(−10). Parentheses around 2a prevent the classic error of dividing only the square-root term by −10.
Interpret the modeled domain
The roots are approximately −0.074 and 4.074. Both solve the polynomial, but negative time lies before the modeled launch. The physically relevant ground time is therefore about 4.074 seconds.

Standard form makes every quadratic coefficient unambiguous

A projectile-height model exposes both useful and extraneous roots

Suppose height in meters is h(t)=−5t²+20t+1.5. To find when the object reaches ground level, solve −5t²+20t+1.5=0 with a=−5, b=20, and c=1.5.

Classify with D
D=20²−4(−5)(1.5)=400+30=430. Because 430 is positive, two distinct real algebraic roots will appear.
Keep the denominator grouped
x=(−20±√430)/(−10). Parentheses around 2a prevent the classic error of dividing only the square-root term by −10.
Interpret the modeled domain
The roots are approximately −0.074 and 4.074. Both solve the polynomial, but negative time lies before the modeled launch. The physically relevant ground time is therefore about 4.074 seconds.

Equivalent quadratic methods reveal different structure

Factoring is fastest when structure is visible

For x²−5x+6=0, (x−2)(x−3)=0 exposes roots immediately. The broader algebra calculator can show this route for supported polynomial input.

Completing the square explains the formula

Rearranging a general quadratic into a squared binomial derives the ± branch and connects roots with vertex form. It is longer numerically but conceptually rich.

The formula handles every valid coefficient set

Root verification catches substitution and sign errors

Rebuild standard form from the original equation and confirm that every coefficient includes its sign.
Check D independently; confusing −b² with (−b)² changes classification.
Substitute each displayed root into ax²+bx+c and confirm that the residual is zero or acceptably close after rounding.
Keep algebraic validity separate from contextual validity: lengths, dates, counts, and elapsed times may exclude a mathematically correct negative root.

Match a, b, and c to ax² + bx + c = 0

Two solutions

Follow the numbers through the formula

Classify with D

Keep the denominator grouped

Interpret the modeled domain

Standard form makes every quadratic coefficient unambiguous

A projectile-height model exposes both useful and extraneous roots

Equivalent quadratic methods reveal different structure

Factoring is fastest when structure is visible

Completing the square explains the formula

The formula handles every valid coefficient set

Root verification catches substitution and sign errors

Important limitations

Match a, b, and c to ax² + bx + c = 0

Two solutions

Follow the numbers through the formula

Classify with D

Keep the denominator grouped

Interpret the modeled domain

Standard form makes every quadratic coefficient unambiguous

A projectile-height model exposes both useful and extraneous roots

Equivalent quadratic methods reveal different structure

Factoring is fastest when structure is visible

Completing the square explains the formula

The formula handles every valid coefficient set

Root verification catches substitution and sign errors