Question

write the quadratic equation whose roots are -5 and -1, and whose leading coefficient is 5. (use the letter x to represent the variable.)

Explanation:

Step1: Start with root form

If $r_1$ and $r_2$ are roots, the equation is $a(x-r_1)(x-r_2)=0$, where $a$ is the leading coefficient. Here, $a=5$, $r_1=-5$, $r_2=-1$.
$5(x-(-5))(x-(-1))=0$
Simplify the factors:
$5(x+5)(x+1)=0$

Step2: Multiply the binomials

First expand $(x+5)(x+1)$ using FOIL:
$(x+5)(x+1)=x^2 + x + 5x + 5 = x^2 + 6x + 5$

Step3: Multiply by leading coefficient

Multiply the quadratic by 5:
$5(x^2 + 6x + 5) = 5x^2 + 30x + 25$

Answer:

$5x^2 + 30x + 25 = 0$