Question

Solve x2 – 8x = 3 by completing the square. Which is the solution set of the equation?

(4 minus StartRoot 19 EndRoot comma 4 + StartRoot 19 EndRoot)
(4 minus StartRoot 11 EndRoot comma 4 + StartRoot 11 EndRoot)
(4 minus StartRoot 8 EndRoot comma 4 + StartRoot 8 EndRoot)
(4 minus StartRoot 3 EndRoot comma 4 + StartRoot 3 EndRoot)

Answer 1

To solve the equation \( x^2 - 8x = 3 \) by completing the square, we follow these steps:

1. Start with the equation:
\[
x^2 - 8x = 3
\]

2. Move 3 to the other side to set the equation to 0:
\[
x^2 - 8x - 3 = 0
\]

3. To complete the square, we need the coefficient of \( x \) to be halved and squared. The coefficient of \( x \) is \(-8\). Half of \(-8\) is \(-4\), and squaring it gives \( 16 \).

4. Add and subtract \( 16 \) inside the equation:
\[
x^2 - 8x + 16 - 16 - 3 = 0
\]
This can be rewritten as:
\[
(x - 4)^2 - 19 = 0
\]

5. Rearranging gives us:
\[
(x - 4)^2 = 19
\]

6. Taking the square root of both sides results in:
\[
x - 4 = \pm \sqrt{19}
\]

7. Solving for \( x \) gives:
\[
x = 4 \pm \sqrt{19}
\]

Thus, the solutions can be presented as:
\[
x = 4 - \sqrt{19} \quad \text{and} \quad x = 4 + \sqrt{19}
\]

The solution set of the equation is:
\[
\{ 4 - \sqrt{19}, 4 + \sqrt{19} \}
\]

So, the correct option from your list is:
\[
(4 - \sqrt{19}, 4 + \sqrt{19})
\]