The quadratic formula provides the solutions to a quadratic equation. Given ax^2 + bx + c = 0, then the solutions are given by:

x=frac(-b±sqrt(b^2-4ac))(2a)

Note that the squared term is calculated once with a positive value of the square root; and once with a negative value. Once the solutions have been found, substitute into the original equation to check there has been no arithmetical errors.

Note also that if b^2 < 4ac then there will be an attempt to find the square root of a negative number, and that therefore there are no real roots.

## Example 1

What, if any, are the solutions for 3x^2-48=0?

Using the formula, remembering that b=0 as there is no coefficient for x:

x=frac(-(0)+-sqrt((0)^2 - 4(3)(-48)))(2(3))

= +-frac(576)(8)

 = +-4

Substitute back to check: 3(4)^2 - 48=0✔ and 3(-4)^2-48=0

Answer: x=+-4

## Example 2

Determine the solutions for 2x^2+18x+28=0.

x=frac(-b ± sqrt(b^2-4ac))(2a)

Substituting:

x=frac(-(18) ± sqrt((18)^2-4(2)(28)))(2(2))

x=frac(-18±sqrt(324 - 224))(4)

x=frac(-18±10)(2)

x=-7 and x=-2

Check:

2(-7)^2+18(-7)+28=0 ✔ and

2(-2)^2+18(-2)+28=0

Answer: x=-2 and x=-7