Question

solve the following equation by factoring.
$z^2 + 2z - 35 = 0$
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 \(-35\) and add up to \(2\). The numbers are \(7\) and \(-5\) since \(7\times(-5)= -35\) and \(7 + (-5)=2\).

Step2: Factor the quadratic

Using the numbers from Step 1, we factor \(z^{2}+2z - 35\) as \((z + 7)(z - 5)\). So the factored form of the equation is \((z + 7)(z - 5)=0\).

Step3: Solve for \(z\)

Set each factor equal to zero:

• For \(z + 7 = 0\), we get \(z=-7\).
• For \(z - 5 = 0\), we get \(z = 5\).

Answer:

The factored form is \((z + 7)(z - 5)=0\), and the solution set is \(\{-7, 5\}\).