Question

finding solutions of an equation
complete the equivalent equation for $-7x - 60 = x^2 + 10x$.
$(x + \diamond)(x + \diamond) = 0$
what are the solutions of $-7x - 60 = x^2 + 10x$?
$x = \diamond$

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:
\(x^2 + 10x + 7x + 60 = 0\)
Simplify the like terms:
\(x^2 + 17x + 60 = 0\)

Step2: Factor the quadratic equation

Now we factor the quadratic \(x^2 + 17x + 60\). We need two numbers that multiply to \(60\) and add up to \(17\). The numbers are \(12\) and \(5\) because \(12\times5 = 60\) and \(12 + 5=17\). So the factored form is:
\((x + 12)(x + 5)=0\)

Step3: Solve for \(x\)

Using the zero - product property, if \((x + 12)(x + 5)=0\), then either \(x+12 = 0\) or \(x + 5=0\).

• For \(x+12=0\), we get \(x=- 12\).
• For \(x + 5=0\), we 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 \(x = - 5\).