Questions: Consider the equation: x^2 - 2x - 8 = 0 A) First, use the "completing the square" process to write this equation in the form (x+D)^2 = E and enter your results below. x^2 - 2x - 8 = 0 is equivalent to: x = -D + sqrt(E) = x = -D - sqrt(E)

Consider the equation: x^2 - 2x - 8 = 0
A) First, use the "completing the square" process to write this equation in the form (x+D)^2 = E and enter your results below.
x^2 - 2x - 8 = 0 is equivalent to:
x = -D + sqrt(E) = x = -D - sqrt(E)

Transcript text: Consider the equation: $x^{2}-2 x-8=0$ A) First, use the "completing the square" process to write this equation in the form $(x+D)^{2}=E$ and enter your results below. $x^{2}-2 x-8=0$ is equivalent to: $x=-D+\sqrt{E}$ $\square$ $=x=-D-\sqrt{E}$ $\square$

Solution

Solution Steps

To solve the quadratic equation $x^2 - 2x - 8 = 0$ by completing the square, we first move the constant term to the other side of the equation. Then, we find the value needed to complete the square for the quadratic and linear terms, add it to both sides, and rewrite the left side as a perfect square trinomial. Finally, we solve for $x$ by taking the square root of both sides.

Step 1: Rewrite the Equation

The given equation is:

\[ x^2 - 2x - 8 = 0 \]

To complete the square, we first need to isolate the constant term on one side of the equation:

\[ x^2 - 2x = 8 \]

Step 2: Complete the Square

To complete the square, we need to add and subtract the square of half the coefficient of $x$. The coefficient of $x$ is $-2$, so half of this is $-1$, and its square is $1$.

Add and subtract $1$ to the left side of the equation:

\[ x^2 - 2x + 1 - 1 = 8 \]

This can be rewritten as:

\[ (x - 1)^2 - 1 = 8 \]

Step 3: Simplify the Equation

Add $1$ to both sides to isolate the perfect square:

\[ (x - 1)^2 = 9 \]

Step 4: Solve for $x$

Now, solve for $x$ by taking the square root of both sides:

\[ x - 1 = \pm \sqrt{9} \]

This gives us:

\[ x - 1 = 3 \quad \text{or} \quad x - 1 = -3 \]