To solve the equation [tex]\( 6(2x + 4)^2 = (2x + 4) + 2 \)[/tex], we will go through the steps in a structured manner. Let's start by simplifying and solving the equation step-by-step.
Firstly, simplify the given equation:
[tex]\[ 6(2x + 4)^2 = (2x + 4) + 2 \][/tex]
To make it easier, let's set [tex]\( u = 2x + 4 \)[/tex], transforming the equation into:
[tex]\[ 6u^2 = u + 2 \][/tex]
Rearrange the equation to set it to zero:
[tex]\[ 6u^2 - u - 2 = 0 \][/tex]
This is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 6 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -2 \)[/tex].
Now, solve the quadratic equation using the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ u = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(6)(-2)}}{2(6)} \][/tex]
[tex]\[ u = \frac{1 \pm \sqrt{1 + 48}}{12} \][/tex]
[tex]\[ u = \frac{1 \pm \sqrt{49}}{12} \][/tex]
[tex]\[ u = \frac{1 \pm 7}{12} \][/tex]
This gives us two possible solutions for [tex]\( u \)[/tex]:
[tex]\[ u_1 = \frac{1 + 7}{12} = \frac{8}{12} = \frac{2}{3} \][/tex]
[tex]\[ u_2 = \frac{1 - 7}{12} = \frac{-6}{12} = -\frac{1}{2} \][/tex]
Recall that [tex]\( u = 2x + 4 \)[/tex]. So we now have two equations to solve for [tex]\( x \)[/tex]:
For [tex]\( u_1 = \frac{2}{3} \)[/tex]:
[tex]\[ 2x + 4 = \frac{2}{3} \][/tex]
[tex]\[ 2x = \frac{2}{3} - 4 \][/tex]
[tex]\[ 2x = \frac{2}{3} - \frac{12}{3} \][/tex]
[tex]\[ 2x = \frac{2 - 12}{3} \][/tex]
[tex]\[ 2x = \frac{-10}{3} \][/tex]
[tex]\[ x = \frac{-10}{3} \cdot \frac{1}{2} \][/tex]
[tex]\[ x = \frac{-10}{6} \][/tex]
[tex]\[ x = -\frac{5}{3} \][/tex]
For [tex]\( u_2 = -\frac{1}{2} \)[/tex]:
[tex]\[ 2x + 4 = -\frac{1}{2} \][/tex]
[tex]\[ 2x = -\frac{1}{2} - 4 \][/tex]
[tex]\[ 2x = -\frac{1}{2} - \frac{8}{2} \][/tex]
[tex]\[ 2x = -\frac{9}{2} \][/tex]
[tex]\[ x = -\frac{9}{4} \][/tex]
Therefore, the solutions to the equation [tex]\( 6(2x + 4)^2 = (2x + 4) + 2 \)[/tex] are:
[tex]\[ x = -\frac{5}{3} \text{ or } x = -\frac{9}{4} \][/tex]
These solutions match the first option, [tex]\( x = -\frac{5}{3} \text{ or } x = -\frac{9}{4} \)[/tex].
