Question

solve the following equation by factoring.
$z^2 + 4z - 21 = 0$

rewrite the equation in a completely factored form.
$\square = 0$
(type your answer in factored form.)

the solution set is $\square$.
(simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(-21\) and add up to \(4\). The numbers are \(7\) and \(-3\) since \(7\times(-3)= -21\) and \(7 + (-3)=4\).

Step2: Factor the quadratic

Using the numbers from Step 1, we can factor the quadratic equation \(z^{2}+4z - 21\) as \((z + 7)(z - 3)\). So the factored form of the equation \(z^{2}+4z - 21 = 0\) is \((z + 7)(z - 3)=0\).

Step3: Solve for \(z\)

Set each factor equal to zero:

• For \(z + 7 = 0\), we get \(z=-7\).
• For \(z - 3 = 0\), we get \(z = 3\).

Answer:

The completely factored form is \((z + 7)(z - 3)=0\). The solution set is \(\{-7, 3\}\).