HexMathv1.0.4

Quadratic equation solver with the quadratic formula

This quadratic equation solver finds the roots of any equation in the form a x squared plus b x plus c equals zero. Type the equation or scan it. HexMath works out the discriminant, uses the quadratic formula or factors when it can, and shows each step to the roots, including a repeated root.

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What this page covers

  • Equations in the form ax² + bx + c = 0, and equations that rearrange into it.
  • The discriminant: two real roots, one repeated root, or no real roots.
  • The quadratic formula, factoring and completing the square.
  • Not covered here: equations of degree three or higher. Use the polynomial calculator, or for any other equation the equation solver.

How to enter the problem

  • Type it or paste it, for example x^2 + 4x + 1 = 0.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1: Solve x² + 4x + 1 = 0.

Problem

Solve x2+4x+1=0\displaystyle x^2 + 4x + 1 = 0.

Answer

Verified

The answer is

x=−2+3x = -2 + \sqrt{3} or x=−2−3x = -2 - \sqrt{3}

Explanation

  1. Read off a, b and c
    a=1a = 1, b=4b = 4, c=1c = 1.
  2. Find the discriminant
    b2−4ac=16−4=12b^2 - 4ac = 16 - 4 = 12, which is positive, so there are two real roots.
  3. Apply the quadratic formula
    x=−b±b2−4ac2a=−4±122x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-4 \pm \sqrt{12}}{2}
  4. Simplify
    12=23\sqrt{12} = 2\sqrt{3}, so x=−4±232=−2±3x = \frac{-4 \pm 2\sqrt{3}}{2} = -2 \pm \sqrt{3}.

Example 2: Solve 2x² + 3x - 2 = 0.

Problem

Solve 2x2+3x−2=0\displaystyle 2x^2 + 3x - 2 = 0.

Answer

Verified

The answer is

x=12x = \frac{1}{2} or x=−2x = -2

Explanation

  1. Find the discriminant
    b2−4ac=9+16=25b^2 - 4ac = 9 + 16 = 25, a perfect square, so the roots are rational.
  2. Apply the quadratic formula
    x=−3±54x = \frac{-3 \pm 5}{4}
  3. Take each sign
    x=24=12x = \frac{2}{4} = \frac{1}{2} or x=−84=−2x = \frac{-8}{4} = -2.

Example 3: Solve 4x² - 12x + 9 = 0.

Problem

Solve 4x2−12x+9=0\displaystyle 4x^2 - 12x + 9 = 0.

Answer

Verified

The answer is

x=32x = \frac{3}{2} (repeated root)

Explanation

  1. Find the discriminant
    b2−4ac=144−144=0b^2 - 4ac = 144 - 144 = 0, so there is one repeated root.
  2. Spot the perfect square
    4x2−12x+9=(2x−3)24x^2 - 12x + 9 = (2x - 3)^2
  3. Solve
    2x−3=02x - 3 = 0, so x=32x = \frac{3}{2}.

Common mistakes

  • Losing the minus sign in -b. When b is negative, -b is positive.
  • Dividing only the square root by 2a. The whole top, -b ± √(b² − 4ac), goes over 2a.
  • Using the formula before moving every term to one side, so c is wrong.
  • Keeping only the + sign. Most quadratics have two roots.

Checks, assumptions and limits

  • Check by substitution. In the second worked example, 2 times one half squared, plus 3 times one half, minus 2, is 0.
  • Check with the sum and product of the roots: they should add to -b/a and multiply to c/a. For the first example they add to -4 and multiply to 1.
  • HexMath can make mistakes. Double check important steps.

Frequently asked questions

For a x squared plus b x plus c equals 0, the roots are minus b, plus or minus the square root of b squared minus 4 a c, all divided by 2 a.

The discriminant is b squared minus 4 a c. Positive means two real roots, zero means one repeated root, and negative means no real roots.

Factor when the numbers make it quick. The formula always works, so use it when factoring is not obvious.

Move every term to one side first, for example x squared equals 3x plus 4 becomes x squared minus 3x minus 4 equals 0. HexMath does this step for you.

Related tasks

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