Quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Uses of the quadratic formula

The main use is for solving the roots of a quadratic equation (finding the values of x where y=0)

It also has another use - the discriminant (the part under the square root) can tell us a lot, for example, the number of roots the equation has. See the page on the discriminant for more information.

Finding the roots

To find the roots of a quadratic equation using the quadratic formula, we substitute the coefficients of the equation into the quadratic formula.

The equation should be in the form ax^2 + bx + c = 0, so just substitute those values into the formula above.

Remember to find both the positive and negative square roots of the discriminant (b^2 - 4ac) to get both roots! You can do this on your calculator by calculating +\sqrt{b^2 - 4ac} and -\sqrt{b^2 - 4ac} separately.

Example questions

Find the roots of 2x^2 + 4x - 6 = 0

• a=2
• b=4
• c=-6
• Substitute into the quadratic formula:
  • x = \frac{-4 \pm \sqrt{4^2 - 4 \times 2 \times -6}}{2 \times 2}
  • x = \frac{-4 \pm \sqrt{16 + 48}}{4}
  • x = \frac{-4 \pm \sqrt{64}}{4}
• Calculate both roots:
  • x = \frac{-4 + 8}{4} = 1
  • x = \frac{-4 - 8}{4} = -3
• Answer: x = 1 and x = -3

Find the roots of x^2 - 3x + 2 = 0

• a=1
• b=-3
• c=2
• Substitute into the quadratic formula:
  • x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \times 1 \times 2}}{2 \times 1}
  • x = \frac{3 \pm \sqrt{9 - 8}}{2}
  • x = \frac{3 \pm \sqrt{1}}{2}
• Calculate both roots:
  • x = \frac{3 + 1}{2} = 2
  • x = \frac{3 - 1}{2} = 1
• Answer: x = 2 and x = 1

Find the roots of 3x^2 + 2x - 8 = 0

• a=3
• b=2
• c=-8
• Substitute into the quadratic formula:
  • x = \frac{-2 \pm \sqrt{2^2 - 4 \times 3 \times -8}}{2 \times 3}
  • x = \frac{-2 \pm \sqrt{4 + 96}}{6}
  • x = \frac{-2 \pm \sqrt{100}}{6}
• Calculate both roots:
  • x = \frac{-2 + 10}{6} = \frac{8}{6} = \frac{4}{3}
  • x = \frac{-2 - 10}{6} = \frac{-12}{6} = -2
• Answer: x = \frac{4}{3} and x = -2