Sure, let's go through the step-by-step solution to solve the equation [tex]\(8x^2 - 48x = -104\)[/tex].
1. Normalize the Quadratic Equation:
We start by rewriting the given quadratic equation so that the coefficient of [tex]\(x^2\)[/tex] is 1. To do this, we will divide every term in the equation by 8:
[tex]\[
\frac{8x^2 - 48x}{8} = \frac{-104}{8}
\][/tex]
Simplifying this, we get:
[tex]\[
x^2 - 6x = -13
\][/tex]
2. Completing the Square:
Next, we need to complete the square on the left-hand side of the equation. To do that, we take the coefficient of [tex]\(x\)[/tex], divide it by 2, and then square it. The coefficient of [tex]\(x\)[/tex] here is -6:
[tex]\[
\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9
\][/tex]
We add and subtract this value on the left-hand side:
[tex]\[
x^2 - 6x + 9 - 9 = -13
\][/tex]
[tex]\[
x^2 - 6x + 9 = -13 + 9
\][/tex]
3. Simplify the Equation:
Now simplify both sides:
[tex]\[
x^2 - 6x + 9 = -4
\][/tex]
4. Factor the Completed Square:
The left-hand side is now a perfect square trinomial, which can be factored as:
[tex]\[
(x - 3)^2 = -4
\][/tex]
5. Solve for x:
To solve for [tex]\(x\)[/tex], we need to take the square root of both sides. Recall that taking the square root of a negative number will introduce the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
x - 3 = \pm \sqrt{-4}
\][/tex]
[tex]\[
x - 3 = \pm 2i
\][/tex]
6. Isolate x:
Finally, we solve for [tex]\(x\)[/tex] by adding 3 to both sides of the equation:
[tex]\[
x = 3 \pm 2i
\][/tex]
Therefore, the solutions to the equation [tex]\(8x^2 - 48x = -104\)[/tex] are:
[tex]\[
x = 3 + 2i \quad \text{and} \quad x = 3 - 2i
\][/tex]
