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Quadratic equations

Use the quadratic formula

Identify a, b, and c from standard form, calculate the discriminant, and keep the roots exact until the final step.

Enter the coefficients

For ax2+bx+c=0ax^2+bx+c=0, enter a nonzero value for a.

Equation2x^2-3x-2=0
Try an example

Result

The discriminant is 25, a positive perfect square, so the equation has two distinct rational roots.

x∈{2,−12}x\in\left\{2,-\frac{1}{2}\right\}
Conditions
  • The equation is written or reducible to ax^2+bx+c=0.
  • The leading coefficient a is not zero.
  • Complex roots are written using i when the discriminant is negative.

Steps

  1. Identify the coefficients Match the equation with ax squared plus bx plus c equals zero.a=2,b=−3,c=−2a=2,\qquad b=-3,\qquad c=-2
  2. Calculate the discriminant Substitute the coefficients into b squared minus 4ac.Δ=(−3)2−4(2)(−2)=25\Delta=(-3)^2-4(2)(-2)=25
  3. Apply the formula Use negative b in the numerator and 2a in the denominator.x=3±254x=\frac{3\pm\sqrt{25}}{4}
  4. Simplify both roots Evaluate the plus and minus cases separately.x=2orx=−12x=2\quad\text{or}\quad x=-\frac{1}{2}
Independent check

Substitution gives 2(2)^2-3(2)-2=0 and 2(-1/2)^2-3(-1/2)-2=0.

Scope

What this quadratic-formula covers

For ax squared plus bx plus c equals zero with a not equal to zero, the roots are x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. The discriminant determines whether the roots are real, repeated, or complex.

Two distinct real roots

A positive discriminant produces two different real values.

Examples: x^2-5x+6=0, 3x^2-2x-1=0

One repeated root

A zero discriminant makes the plus and minus branches identical.

Examples: x^2+4x+4=0, 4x^2-12x+9=0

Complex conjugate roots

A negative discriminant produces a conjugate pair involving i.

Examples: x^2+1=0, x^2+2x+5=0

Exact irrational roots

A positive non-square discriminant remains in simplified radical form before any approximation.

Examples: x^2-2x-1=0, 3x^2+6x-2=0

How to use it

Enter enough information for one clear task

  1. 1
    Move every term to one side

    Rewrite the equation as ax squared plus bx plus c equals zero and combine like terms.

  2. 2
    Record signed coefficients

    The sign belongs to the coefficient. For 2x squared minus 3x minus 2, b is negative 3 and c is negative 2.

  3. 3
    Use the discriminant first

    Calculate b squared minus 4ac to predict the number and type of roots before simplifying the formula.

  4. 4
    Check exact roots

    Substitute each root into the original quadratic or verify that their sum is negative b over a and their product is c over a.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Two integer roots

The discriminant is 1, so simplify both formula branches.

x2−5x+6=0x^2-5x+6=0

Expected result

x∈{2,3}x\in\{2,3\}

Repeated root

The discriminant is zero.

x2+4x+4=0x^2+4x+4=0

Expected result

x=−2x=-2

Complex roots

The discriminant is negative 4, so use the principal square root with i.

x2+1=0x^2+1=0

Expected result

x=±ix=\pm i

Unequal leading coefficient

Substitute a=3, b=-2, and c=-1.

3x2−2x−1=03x^2-2x-1=0

Expected result

x∈{1,−13}x\in\left\{1,-\frac{1}{3}\right\}

Irrational roots

Simplify the square root of the discriminant but keep the result exact.

x2−2x−1=0x^2-2x-1=0

Expected result

x=1±2x=1\pm\sqrt{2}

Two rational roots

The discriminant is 64, then divide both numerators by 8.

4x2+4x−3=04x^2+4x-3=0

Expected result

x∈{12,−32}x\in\left\{\frac{1}{2},-\frac{3}{2}\right\}

Complete example

Simplify a non-square discriminant exactly

This quadratic does not factor into integer binomials. The formula gives exact radical roots and makes their symmetry visible.

3x2+6x−2=03x^2+6x-2=0
  1. 1
    Identify a, b, and c

    Read each signed coefficient from standard form.

    a=3,b=6,c=−2a=3,\qquad b=6,\qquad c=-2
  2. 2
    Compute the discriminant

    The negative constant makes the second term add to b squared.

    Δ=62−4(3)(−2)=60\Delta=6^2-4(3)(-2)=60
  3. 3
    Substitute into the formula

    Use 2a=6 as the denominator of both branches.

    x=−6±606x=\frac{-6\pm\sqrt{60}}{6}
  4. 4
    Simplify the radical and fraction

    Since the square root of 60 is 2 times the square root of 15, divide every numerator term by 6.

    x=−1±153x=-1\pm\frac{\sqrt{15}}{3}
x=−1±153x=-1\pm\frac{\sqrt{15}}{3}

Verification: The roots sum to -2=-b/a and multiply to -2/3=c/a, which independently checks the exact pair.

Avoidable errors

Common mistakes and how to fix them

Losing the sign of b

Problem: For 2x^2-3x-2=0, use b=3.

Why it matters: The coefficient includes its sign, so b is negative 3.

Better approach: Write a, b, and c on a separate line before substituting.

Dividing only the radical by 2a

Problem: x=-b+sqrt(discriminant)/(2a).

Why it matters: The entire numerator, including negative b and the radical, is divided by 2a.

Better approach: Keep a fraction bar under the complete numerator until both branches are simplified.

Interpreting a negative discriminant as no answer

Problem: Stop when the discriminant is below zero.

Why it matters: There are no real roots, but there are two complex conjugate roots.

Better approach: State the number system and write the square root using i when complex answers are allowed.

Rounding before simplifying

Problem: Replace a radical with a short decimal in the middle of the calculation.

Why it matters: Early rounding hides exact structure and can accumulate error.

Better approach: Keep the radical exact, then add a labeled decimal approximation only if it is useful.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Each root is substituted into the original quadratic when practical.
  • The pair is checked against root sum -b/a and root product c/a.
  • The discriminant classification agrees with the reported root type.
  • Radicals and fractions are simplified before optional decimal approximation.

When to stop and revise the input

  • If a equals zero, the equation is linear and the quadratic formula does not apply.
  • An equation must first be reduced to standard form with all like terms combined.
  • Decimal coefficients may require exact rational interpretation or a stated rounding rule.
  • A graph can support interpretation but does not replace algebraic root verification.

Common questions

Use the quadratic formula FAQ

What is the discriminant?

The discriminant is b squared minus 4ac. Its sign tells whether a real-coefficient quadratic has two distinct real roots, one repeated real root, or two complex conjugate roots.

Does the quadratic formula work when factoring does not?

Yes, for every quadratic with a nonzero leading coefficient. Factoring may be shorter when simple factors are visible, but the quadratic formula remains valid.

Why should I keep radical answers exact?

Exact radicals preserve the true value and make later algebra more reliable. A decimal can be added afterward and labeled as an approximation.

What does a repeated root mean on the graph?

The parabola touches the x-axis at one point instead of crossing it at two distinct points. Algebraically, the discriminant is zero and both formula branches give the same value.

How can I check quadratic roots without repeating the formula?

Substitute each root into the original equation. You can also check that the two roots add to negative b over a and multiply to c over a.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.