Math

Quadratic Formula Calculator

Solve any quadratic equation instantly with our free quadratic formula calculator.

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What is Quadratic Formula Calculator?

A quadratic equation has the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It's called "quadratic" because the highest power of the variable x is 2. Quadratic equations describe parabolas — U-shaped curves — and appear throughout algebra, physics, engineering, and everyday problem-solving, from calculating projectile trajectories to optimizing areas and profits. This calculator uses the quadratic formula, one of the most important results in algebra, to find the exact roots (solutions) of any quadratic equation, whether those roots are real numbers or complex numbers.

When to Use This Calculator

  • Solve any quadratic equation of the form ax² + bx + c = 0 quickly and exactly
  • Determine the number and type of roots from the discriminant before solving
  • Find the vertex and axis of symmetry of a parabola without graphing
  • Check hand-worked algebra homework, exam answers, and practice problems
  • Model projectile motion, area optimization, and profit functions in applied math
  • Prepare for standardized tests such as the SAT, ACT, and GCSE

Steps:

  1. Enter the coefficients a, b, and c from your equation ax² + bx + c = 0.
  2. The calculator computes the discriminant (b² - 4ac) to determine the type of roots.
  3. The quadratic formula is applied to find the exact root values.
  4. Review the full step-by-step solution, plus the vertex and axis of symmetry.

Formula

x = (-b ± √(b² - 4ac)) / 2a Discriminant: Δ = b² - 4ac If Δ > 0: two distinct real roots If Δ = 0: one repeated real root If Δ < 0: two complex conjugate roots Vertex: x = -b/2a, y = c - b²/4a

Use Cases

  • Solving algebra homework and checking your work
  • Finding the trajectory peak and landing point of a projectile in physics
  • Optimizing area, profit, or revenue functions in business and engineering
  • Preparing for standardized tests like the SAT, ACT, or GCSE

Key Benefits

  • Instant, exact roots for any quadratic equation
  • Automatically handles real and complex roots
  • Full step-by-step working, not just the final answer
  • Also shows the vertex and axis of symmetry

Pro Tips

  • If a, b, and c are all divisible by a common factor, simplify the equation first
  • Check your answer by substituting each root back into the original equation
  • A quick way to estimate real-root sign: if a and c have opposite signs, the discriminant is guaranteed positive
  • The sum of the roots is always -b/a, and the product is always c/a — useful for a fast sanity check

Common Mistakes to Avoid

  • Forgetting the ± sign, which produces only one of the two roots
  • Mixing up the signs of b and c when substituting into the formula
  • Assuming a negative discriminant means "no solution" instead of "complex solutions"
  • Dividing by 2a incorrectly when a is negative

Key Terms Explained

Discriminant: The expression b² - 4ac, which determines the nature of the roots
Root: A value of x that satisfies the equation, also called a zero or solution
Vertex: The turning point of the parabola, either its minimum or maximum
Axis of Symmetry: The vertical line x = -b/2a that the parabola is mirrored across

Related Concepts

Example

For x² - 3x + 2 = 0 (a=1, b=-3, c=2): the discriminant is (-3)² - 4(1)(2) = 9 - 8 = 1. Since Δ > 0, there are two real roots: x = (3 ± 1) / 2, giving x₁ = 2 and x₂ = 1.

Interpreting Your Results

The discriminant Δ = b² - 4ac is the key result. When Δ > 0 the parabola crosses the x-axis twice and there are two distinct real roots. When Δ = 0 the parabola just touches the axis at one point, giving one repeated real root. When Δ < 0 there are no real intersections, only two complex conjugate roots. The roots themselves are the x-values where the parabola crosses the axis, and the vertex, found at x = -b/2a, sits exactly on the axis of symmetry halfway between the two real roots.

Frequently Asked Questions

What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves any equation of the form ax² + bx + c = 0 by giving both roots directly from the coefficients a, b, and c.
What does a negative discriminant mean?
A negative discriminant (b² - 4ac < 0) means the equation has no real roots — the parabola never touches the x-axis. Instead, it has two complex conjugate roots of the form p ± qi.
Can a equal zero in a quadratic equation?
No. If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0.
How do I find the vertex of a parabola from a, b, and c?
The vertex x-coordinate is -b/2a. Substitute that value back into the original equation to get the corresponding y-coordinate, c - b²/4a.
How do I solve a quadratic equation by completing the square?
Move the constant to the right, add the square of half the coefficient of x to both sides, then take the square root. For x² + 6x + 8 = 0, rewrite as x² + 6x = -8, add (6/2)² = 9 to both sides to get x² + 6x + 9 = 1, which factors as (x + 3)² = 1. Taking the square root gives x + 3 = ±1, so x = -3 ± 1, meaning x = -2 and x = -4.
How do I solve a quadratic equation by factoring?
Factor the quadratic into two binomials and set each factor to zero. For x² - 5x + 6 = 0, look for two numbers that multiply to 6 and add to -5, which are -2 and -3, so it factors as (x - 2)(x - 3) = 0. Setting each factor to zero gives x - 2 = 0 or x - 3 = 0, so the roots are x = 2 and x = 3. Factoring works only when the roots are rational; otherwise use the quadratic formula.
What is the axis of symmetry of a parabola?
The axis of symmetry is the vertical line x = -b/2a that splits the parabola into two mirror images, and it always passes through the vertex. For x² - 4x + 3 = 0, with a = 1 and b = -4, the axis is x = -(-4)/(2 × 1) = 2. The two real roots of an equation always sit at equal distances on either side of this line.
What do complex roots look like?
When the discriminant is negative, the roots include the imaginary unit i, where i² = -1. For x² + 1 = 0, the formula gives x = ±√(-1), written as x = i and x = -i. Every quadratic with real coefficients has its complex roots in conjugate pairs, so if a + bi is a root, then a - bi is the other, and their sum is 2a and their product is a² + b².
How do I solve a quadratic equation with fraction or decimal coefficients?
Multiply the whole equation by the common denominator to clear the fractions first, then solve normally. For (1/2)x² - x + 1/2 = 0, multiply every term by 2 to get x² - 2x + 1 = 0, which factors as (x - 1)² = 0. The equation then has the repeated root x = 1. Clearing fractions avoids mistakes and keeps the coefficients as simple integers.
How do I solve a difference of squares equation?
A difference of squares a² - b² factors as (a - b)(a + b). For x² - 9 = 0, treat x² as a² and 9 as b² with b = 3, so it factors as (x - 3)(x + 3) = 0. Setting each factor to zero gives x - 3 = 0 or x + 3 = 0, so the roots are x = 3 and x = -3. This pattern solves instantly without the quadratic formula.
Which method should I choose: factoring, completing the square, or the formula?
Try factoring first when the roots look rational and the numbers are small, since it is fastest. Use completing the square when the equation has a leading coefficient of 1 and you also need to rewrite it in vertex form. Use the quadratic formula for everything else, because it always works, including for irrational and complex roots. All three methods give identical answers when applied correctly.

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