Respuesta :
9 years
Explanation
Step 1
set the equations.
let
x represents the lisa's age
y represents the twin bother's age
so
a)the product of the ages of Lisa and her twin bothers is 36
hence
[tex]xy=36\Rightarrow equation(1)[/tex]b)and the sum of their ages is 13
[tex]x+y=13\Rightarrow equation(2)[/tex]Step 2
solve the equations
a) isolate the y value in equation (2) then replace in eqaution (1)
[tex]\begin{gathered} x+y=13\Rightarrow equation(2) \\ \text{subtract x in both sides} \\ x+y-x=13-x \\ y=13-x \end{gathered}[/tex]now , replace in eq(1)
[tex]\begin{gathered} xy=36\Rightarrow equation(1) \\ x(13-x)=36 \\ 13x-x^2=36 \\ \text{subtract 36 in both sides an reorder} \\ 13x-x^2-36=36-36 \\ -x^2+13x-36=0 \end{gathered}[/tex]we need to solve this quadratic equation, let's use the quadratic formula
[tex]\begin{gathered} \text{for ax}^2+bx+c=0 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \end{gathered}[/tex]hence
[tex]\begin{gathered} -x^2+13x-36=0\Rightarrow ax^2+bx+c \\ so \\ a=-1 \\ b=13 \\ c=-36 \end{gathered}[/tex]now, to find the solutino for x, let's replace in the formula
[tex]\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-(13)\pm\sqrt[]{13^2-4(-1)(-36)}}{2(-1)} \\ x=\frac{-(13)\pm\sqrt[]{169-144}}{-2} \\ x=\frac{-(13)\pm\sqrt[]{25}}{-2} \\ x=\frac{-(13)\pm5}{-2} \end{gathered}[/tex]we have the symbol
[tex]\pm[/tex]it means, there are two solutions, let's check
[tex]\begin{gathered} x=\frac{-(13)\pm5}{-2} \\ x_1=\frac{-(13)+5}{-2}=\frac{-8}{-2}=4 \\ x_2=\frac{-(13)-5}{-2}=\frac{-18}{-2}=9 \end{gathered}[/tex]so,
[tex]-x^2+13x-36=0=(x-4)(x-9)[/tex]Step 3
let's solve the quadratic equation by factoring
[tex]\begin{gathered} -x^2+13x-36=0 \\ \text{change the signs} \\ x^2-13x+36=0 \\ \text{rewrite -13x as -4x-9x} \\ x^2-4x-9x+36=0 \\ \text{factorize} \\ x(x-4)-9(x-4)=0 \\ (x-4)(x-9)=0 \end{gathered}[/tex]so, the posibles values for Lisa´s age are
9 or 4
as we don't know who is older, the ages are 4 and 9
let's prove
a)the product of the ages of Lisa and her twin bothers is 36
[tex]9\cdot4=36[/tex]b)
and the sum of their ages is 13.
[tex]9+4=13[/tex]I hope this helps you
x
