Question

finding solutions of an equation
complete the equivalent equation for -7x - 60 = x² + 10x.
(x + )(x + ) = 0
what are the solutions of -7x - 60 = x² + 10x?
-5
-3
3
5

Explanation:

Step1: Rearrange the equation

First, we need to rearrange the given equation \(-7x - 60 = x^{2}+10x\) into standard quadratic form \(ax^{2}+bx + c = 0\).
We move all terms to one side:
\(x^{2}+10x + 7x+60 = 0\)
Combining like terms, we get:
\(x^{2}+17x + 60 = 0\)

Step2: Factor the quadratic

Now we factor the quadratic expression \(x^{2}+17x + 60\). We need two numbers that multiply to \(60\) and add up to \(17\). The numbers \(12\) and \(5\) work because \(12\times5 = 60\) and \(12 + 5=17\).
So, \(x^{2}+17x + 60=(x + 12)(x + 5)=0\)

Step3: Find the solutions

To find the solutions, we set each factor equal to zero:

• For \(x + 12 = 0\), we solve for \(x\) and get \(x=-12\)
• For \(x + 5 = 0\), we solve for \(x\) and get \(x=-5\)

Answer:

The equivalent equation is \((x + 12)(x + 5)=0\). The solutions of the equation \(-7x - 60 = x^{2}+10x\) are \(x=-12\) and \(x = - 5\). Among the given options, the solution from the options is \(-5\) (and the other solution \(-12\) is not in the given options, but based on the options provided, the correct one from the list is \(-5\)). So the answer for the solution from the given options is \(-5\) (and the factored form has \(12\) and \(5\) in the blanks).

For the factored form: \((x + 12)(x + 5)=0\) (so the blanks are \(12\) and \(5\) in order). For the solution from the given options, the answer is \(-5\).