Question

1. $x^2 = -10x - 16$

Explanation:

Step1: Rearrange the equation to standard quadratic form

We start with the equation \(x^{2}=-10x - 16\). To write it in the standard quadratic form \(ax^{2}+bx + c = 0\), we add \(10x\) and \(16\) to both sides of the equation.
\[x^{2}+10x + 16=0\]

Step2: Factor the quadratic equation

We need to find two numbers that multiply to \(16\) (the constant term \(c\)) and add up to \(10\) (the coefficient of the \(x\) term \(b\)). The numbers \(2\) and \(8\) satisfy this condition because \(2\times8 = 16\) and \(2 + 8=10\). So we can factor the quadratic as:
\[(x + 2)(x+8)=0\]

Step3: Solve for \(x\) using the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So, for \((x + 2)(x + 8)=0\), we set each factor equal to zero and solve for \(x\):

• If \(x+2=0\), then subtracting \(2\) from both sides gives us \(x=-2\).
• If \(x + 8=0\), then subtracting \(8\) from both sides gives us \(x=-8\).

Answer:

\(x=-2\) or \(x = - 8\)