To solve the quadratic equation [tex]\(-27 = 4x^2 - 24x\)[/tex] by completing the square, let's go through the steps meticulously:
1. Start with the given equation:
[tex]\[
-27 = 4x^2 - 24x
\][/tex]
2. Factor out the coefficient of [tex]\(x^2\)[/tex] from the right-hand side:
[tex]\[
-27 = 4 (x^2 - 6x)
\][/tex]
3. We need to complete the square inside the parenthesis. To do so, we add and subtract the square of half the coefficient of [tex]\(x\)[/tex]:
[tex]\[
-27 + \square = 4 (x^2 - 6x + 9)
\][/tex]
4. The term that was added inside the parenthesis is:
[tex]\[
(6/2)^2 = 9
\][/tex]
Thus, we need to add [tex]\(4 \times 9 = 36\)[/tex] to the left-hand side to keep the equation balanced:
[tex]\[
-27 + 36 = 4 (x^2 - 6x + 9)
\][/tex]
So the missing piece is 36.
5. Simplifying the left-hand side, we get:
[tex]\[
9 = 4 (x - 3)^2
\][/tex]
6. Divide both sides by 4 to isolate the squared term:
[tex]\[
\frac{9}{4} = (x - 3)^2
\][/tex]
7. Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
\pm \sqrt{\frac{9}{4}} = x - 3
\][/tex]
8. Simplify the square root:
[tex]\[
\pm \frac{3}{2} = x - 3
\][/tex]
9. Solving for [tex]\(x\)[/tex], we get two solutions:
[tex]\[
x = 3 + \frac{3}{2}
\][/tex]
[tex]\[
x = 3 - \frac{3}{2}
\][/tex]
10. Simplifying the expressions, we obtain:
[tex]\[
x = 5.121320343559642
\][/tex]
[tex]\[
x = 0.8786796564403576
\][/tex]
Thus, the missing pieces and solutions are as follows:
- The value added to both sides to complete the square inside the parenthesis is [tex]\(9\)[/tex].
- The solutions to the equation are approximately [tex]\(x = 5.121320343559642\)[/tex] and [tex]\(x = 0.8786796564403576\)[/tex].
