Let's solve the equation [tex]\( 6(2x + 4)^2 = (2x + 4) + 2 \)[/tex] step by step.
1. Expand and simplify the equation:
Rewrite the given equation:
[tex]\[
6(2x + 4)^2 = (2x + 4) + 2
\][/tex]
2. Multiply out the left-hand side:
[tex]\( 6(2x + 4)^2 \)[/tex] involves squaring [tex]\( 2x+4 \)[/tex] first:
[tex]\[
(2x + 4)^2 = (2x + 4)(2x + 4) = 4x^2 + 16x + 16
\][/tex]
Now, multiply by 6:
[tex]\[
6(4x^2 + 16x + 16) = 24x^2 + 96x + 96
\][/tex]
3. Combine terms on both sides:
Substitute this back into the equation:
[tex]\[
24x^2 + 96x + 96 = (2x + 4) + 2
\][/tex]
Simplify the right-hand side:
[tex]\[
2x + 4 + 2 = 2x + 6
\][/tex]
Hence, the equation is:
[tex]\[
24x^2 + 96x + 96 = 2x + 6
\][/tex]
4. Move all terms to one side to set the equation to 0:
[tex]\[
24x^2 + 96x + 96 - 2x - 6 = 0
\][/tex]
Combine like terms:
[tex]\[
24x^2 + 94x + 90 = 0
\][/tex]
5. Solve the quadratic equation:
To solve [tex]\( 24x^2 + 94x + 90 = 0 \)[/tex], we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 24 \)[/tex], [tex]\( b = 94 \)[/tex], and [tex]\( c = 90 \)[/tex].
First, compute the discriminant:
[tex]\[
b^2 - 4ac = 94^2 - 4 \times 24 \times 90
\][/tex]
Calculating each term:
[tex]\[
94^2 = 8836
\][/tex]
[tex]\[
4 \times 24 \times 90 = 8640
\][/tex]
Therefore:
[tex]\[
8836 - 8640 = 196
\][/tex]
6. Find the square root of the discriminant:
[tex]\[
\sqrt{196} = 14
\][/tex]
7. Substitute into the quadratic formula:
[tex]\[
x = \frac{-94 \pm 14}{48}
\][/tex]
Calculate the two possible solutions:
For [tex]\( x = \frac{-94 + 14}{48} = \frac{-80}{48} = -\frac{5}{3} \)[/tex]
For [tex]\( x = \frac{-94 - 14}{48} = \frac{-108}{48} = -\frac{9}{4} \)[/tex]
Therefore, the solutions to the equation [tex]\( 6(2 x+4)^2 = (2 x+4) + 2 \)[/tex] are:
[tex]\[
x = -\frac{5}{3} \quad \text{and} \quad x = -\frac{9}{4}
\][/tex]
