Question

1. $x^2 + 6x = 7$

Explanation:

Step1: Rewrite the equation in standard form

We start with the equation \(x^{2}+6x = 7\). To write it in the standard quadratic form \(ax^{2}+bx + c=0\), we subtract 7 from both sides of the equation.
\(x^{2}+6x - 7=0\)

Step2: Factor the quadratic equation

We need to find two numbers that multiply to - 7 and add up to 6. The numbers are 7 and - 1 since \(7\times(-1)=-7\) and \(7+( - 1)=6\).
So we can factor the quadratic as \((x + 7)(x-1)=0\)

Step3: Solve for x using the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\) (or both).
For \((x + 7)(x - 1)=0\), we set each factor equal to zero:

• If \(x+7 = 0\), then \(x=-7\)
• If \(x - 1=0\), then \(x = 1\)

Answer:

\(x=-7\) or \(x = 1\)