Question

10 fill in the blank 2.5 points find the zeros/roots by completing the square. $x^2 + 2x - 10 = 53$ write answers as {x,x} with no spaces. press shift to create }. for example, if your answer is x = 2 and x = -5, your answer would be written as {2,-5} type your answer...

Explanation:

Step1: Isolate the quadratic and linear terms

First, we need to isolate the \(x^2\) and \(x\) terms on one side of the equation. We start with the equation \(x^2 + 2x - 10 = 53\). Add 10 to both sides to get:
\(x^2 + 2x = 53 + 10\)
\(x^2 + 2x = 63\)

Step2: Complete the square

To complete the square for the expression \(x^2 + 2x\), we take half of the coefficient of \(x\) (which is 2), square it, and add it to both sides. Half of 2 is 1, and squaring 1 gives 1. So we add 1 to both sides:
\(x^2 + 2x + 1 = 63 + 1\)
The left side is now a perfect square trinomial, which factors as \((x + 1)^2\). So we have:
\((x + 1)^2 = 64\)

Step3: Solve for \(x\)

Take the square root of both sides. Remember that when we take the square root of a number, we get both a positive and a negative root:
\(x + 1 = \pm\sqrt{64}\)
\(\sqrt{64} = 8\), so:
\(x + 1 = \pm 8\)

Now, solve for \(x\) in both cases:

• Case 1: \(x + 1 = 8\)

Subtract 1 from both sides: \(x = 8 - 1 = 7\)

• Case 2: \(x + 1 = -8\)

Subtract 1 from both sides: \(x = -8 - 1 = -9\)

Answer:

{7,-9}