Question

solve: $x^2 + 10x = -9$; options: -1 and -9, 1 and -1, 1 and -12, 1 and -9

Explanation:

Step1: Rearrange the equation

First, we rewrite the given equation \(x^{2}+10x = - 9\) in standard quadratic form \(ax^{2}+bx + c=0\) by adding 9 to both sides. So we get \(x^{2}+10x + 9=0\).

Step2: Factor the quadratic equation

We need to find two numbers that multiply to \(9\) (the value of \(c\)) and add up to \(10\) (the value of \(b\)). The numbers \(1\) and \(9\) satisfy this because \(1\times9 = 9\) and \(1 + 9=10\). So we can factor the quadratic as \((x + 1)(x+9)=0\).

Step3: Solve for x

Using the zero - product property, if \((x + 1)(x + 9)=0\), then either \(x+1 = 0\) or \(x + 9=0\).

• If \(x+1=0\), then \(x=-1\).
• If \(x + 9=0\), then \(x=-9\).

Answer:

-1 and -9