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Quadratic Equation Calculator

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Worked example

Step-by-step solution

2x2−5x−3=02x^2 - 5x - 3 = 0

Step 1 / 4

Identify a, b and c

The equation is already in standard form.

a=2,  b=−5,  c=−3a = 2,\; b = -5,\; c = -3

How to use the Quadratic Equation Calculator

This quadratic equation calculator puts your equation in standard form, checks the discriminant, then solves by factoring when it can and with the quadratic formula when it must, giving exact and decimal roots.

  1. Type the equation, for example 2x² − 5x − 3 = 0 (use ^ for powers: 2x^2).
  2. Press Solve it. The calculator rewrites it as ax² + bx + c = 0 and reads off a, b and c.
  3. Follow the discriminant step to see how many real roots there are.
  4. Use the exact roots for homework and the decimals to check a graph.

The quadratic formula and the discriminant

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Every quadratic equation ax² + bx + c = 0 with a ≠ 0 can be solved with the quadratic formula. The expression under the square root, b² − 4ac, is the discriminant: positive means two real roots, zero means one repeated root, and negative means two complex roots.

When the discriminant is a perfect square the quadratic factors over the integers, and factoring is usually the faster method taught in class. The calculator uses factoring in that case and the formula otherwise.

Frequently asked questions

When should I factor instead of using the quadratic formula?

Factor when the discriminant b² − 4ac is a perfect square; the roots are then rational. Otherwise use the quadratic formula, which always works.

Can the calculator give complex roots?

Yes. When the discriminant is negative the roots are written as a ± bi, for example x² + 2x + 5 = 0 has roots −1 ± 2i.

Does the equation have to equal zero?

No. Enter it in any form, such as x² = 3x + 10; the first step moves every term to one side.

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Have a different kind of math problem? Try the free math solver. Use the subject menu for physics and chemistry.