To find the roots of the function [tex]\( y = 4x^2 + 2x - 30 \)[/tex], we need to set [tex]\( y = 0 \)[/tex]. Thus, we solve the equation:
[tex]\[ 0 = 4x^2 + 2x - 30 \][/tex]
First, let's outline the steps for factoring the equation:
1. Identify the quadratic equation:
[tex]\[ 4x^2 + 2x - 30 = 0 \][/tex]
2. Factor out the greatest common factor (GCF):
We notice that there's no GCF other than 1 that can be factored out from all the terms.
3. Factor the quadratic expression completely:
We look for two numbers that multiply to give the product of the coefficient of [tex]\( x^2 \)[/tex] (which is 4) and the constant term (-30), and that add up to the coefficient of the linear term (2).
However, in this case, we use the quadratic formula directly because factoring by inspection can be challenging.
The quadratic formula is [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -30 \)[/tex].
Applying these values to the quadratic formula:
[tex]\[
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 4 \cdot (-30)}}{2 \cdot 4}
\][/tex]
[tex]\[
x = \frac{-2 \pm \sqrt{4 + 480}}{8}
\][/tex]
[tex]\[
x = \frac{-2 \pm \sqrt{484}}{8}
\][/tex]
[tex]\[
x = \frac{-2 \pm 22}{8}
\][/tex]
This gives us two solutions:
[tex]\[
x = \frac{-2 + 22}{8} = \frac{20}{8} = \frac{5}{2}
\][/tex]
[tex]\[
x = \frac{-2 - 22}{8} = \frac{-24}{8} = -3
\][/tex]
4. Set each factor equal to zero and solve:
The solutions [tex]\( x = \frac{5}{2} \)[/tex] and [tex]\( x = -3 \)[/tex] are the roots of the quadratic equation.
Therefore, the roots of the function [tex]\( 4x^2 + 2x - 30 \)[/tex] are [tex]\( \boxed{-3} \)[/tex] and [tex]\( \boxed{\frac{5}{2}} \)[/tex].
