Question

question 1-1
solve the following polynomial by factoring.
$d^2 - d = 12$

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the equation \(d^{2}-d = 12\) in standard quadratic form \(ax^{2}+bx + c=0\). Subtract 12 from both sides:
\(d^{2}-d - 12=0\)

Step2: Factor the quadratic

We need to find two numbers that multiply to \(- 12\) (the product of \(a = 1\) and \(c=-12\)) and add up to \(-1\) (the coefficient of \(d\)). The numbers are \(-4\) and \(3\) because \((-4)\times3=-12\) and \(-4 + 3=-1\). So we can factor the quadratic as:
\((d - 4)(d+3)=0\)

Step3: Solve for \(d\)

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

• If \(d - 4=0\), then \(d=4\).
• If \(d + 3=0\), then \(d=-3\).

Answer:

The solutions are \(d = 4\) and \(d=-3\)