[tex] \huge \boxed{\mathfrak{Question} \downarrow}[/tex]
For what values of x is the equation x² − x = 20 true?
[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]
[tex] \sf \: x ^ { 2 } - x = 20[/tex]
Subtract 20 from both sides.
[tex] \sf \: x^{2}-x-20=0 [/tex]
To solve the equation, factor [tex]\sf\: x^{2}-x-20[/tex] using formula [tex]\sf\:x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right)[/tex]. To find a and b, set up a system to be solved.
[tex] \sf \: a+b=-1 \\ \sf \: ab=-20 [/tex]
As ab is negative, a and b have the opposite signs. As a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -20.
[tex] \sf1,-20 \\ \sf 2,-10 \\ \sf4,-5 [/tex]
Calculate the sum for each pair.
[tex] \sf1-20=-19 \\ \sf2-10=-8 \\ \sf 4-5=-1 [/tex]
The solution is the pair that gives sum -1.
[tex] \sf \: a=-5 \\ \sf \: b=4 [/tex]
Rewrite factored expression [tex]\sf \left(x+a\right)\left(x+b\right) [/tex] using the obtained values.
[tex] \sf\left(x-5\right)\left(x+4\right) [/tex]
To find equation's solutions, solve x-5=0 and x+4=0.
[tex] \bf \: x=5 \\ \bf x=-4[/tex]
⇨ The values of x are 5 & - 4.
