Question

solve for all values of x by factoring.\\(x^2 - 101 = -1\\)

Explanation:

Step1: Simplify the equation

First, we need to simplify the given equation \(x^{2}-101 = - 1\). Add 101 to both sides of the equation:
\(x^{2}-101 + 101=-1 + 101\)
\(x^{2}=100\)

Step2: Rewrite in factored form

We can rewrite the equation \(x^{2}=100\) as \(x^{2}-100 = 0\). Notice that \(x^{2}-100\) is a difference of squares, and the formula for factoring a difference of squares is \(a^{2}-b^{2}=(a + b)(a - b)\). Here, \(a=x\) and \(b = 10\), so we factor the left - hand side:
\((x + 10)(x - 10)=0\)

Step3: Solve for x

According to the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So we set each factor equal to zero:

• If \(x+10 = 0\), then \(x=-10\).
• If \(x - 10=0\), then \(x = 10\).

Answer:

\(x=-10\) or \(x = 10\)