Using the Quadratic Formula

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 $2x^2+18x+28=0$ .

Using the quadratic formula:

$x=frac(-b ± sqrt(b^2-4ac))(2a)$

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: $x=-2 and x=-7$