Using the Quadratic Formula to find Complex Roots

We want to use the discriminant of the given quadratic equation to determine the number and type of the roots. If we don't want to know the exact values of the roots, we only need to work with the discriminant. From Part A, we know that the discriminant of the given equation is $36 .$ $Equation : Discriminant : 8 x^{2} = - 2 x + 1 36$ Since the discriminant is greater than zero and a perfect square, the quadratic equation has two rational roots.

c

We will use the Quadratic Formula to find the exact solutions of the given equation.

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

Recall that we have already identified the values of

a,

b,

and

c

in Part A, as well as the discriminant,

b^{2} - 4 a c .

a = 8, b = 2, c = - 1 Discriminant : 36

Let's substitute these values into the Quadratic Formula.

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

SubstituteValuesSubstitute values

x = \frac{- 2 \pm 3 6}{2 ( 8 )}

Simplify

CalcRootCalculate root

x = \frac{- 2 \pm 6}{2 ( 8 )}

MultiplyMultiply

x = \frac{- 2 \pm 6}{1 6}

The solutions for this equation are

x = \frac{- 2 \pm 6}{1 6} .

Let's separate them into the positive and negative cases.

$x = \frac{- 2 \pm 6}{1 6}$
$x_{1} = \frac{- 2 + 6}{1 6}$	$x_{2} = \frac{- 2 - 6}{1 6}$
$x_{1} = \frac{4}{1 6}$	$x_{2} = \frac{- 8}{1 6}$
$x_{1} = \frac{1}{4}$	$x_{2} = - \frac{1}{2}$

Using the Quadratic Formula, we found that the solutions of the given equation are $x_{1} = \frac{1}{4}$ and $x_{2} = - \frac{1}{2} .$

Exercise 1.10 - Solution