Answer:
x=-2 solves the equation h(x)=10.
Step-by-step explanation:
Given the function
[tex]h\left(x\right)\:=\:2x^2\:-\:7x-12[/tex]
Putting [tex]h(x)=10[/tex] in the function
[tex]10=2x^2-7x-12[/tex]
switch both sides
[tex]2x^2-7x-12=10[/tex]
add 12 to both sides
[tex]2x^2-7x-12+12=10+12[/tex]
[tex]2x^2-7x=22[/tex]
Divide both sides by 2
[tex]\frac{2x^2-7x}{2}=\frac{22}{2}[/tex]
[tex]x^2-\frac{7x}{2}=11[/tex]
Rewriting in the form x²+2ax+a²
[tex]\mathrm{Add\:}\left(-\frac{7}{4}\right)^2\mathrm{\:to\:both\:sides}[/tex]
[tex]x^2-\frac{7x}{2}+\left(-\frac{7}{4}\right)^2=11+\left(-\frac{7}{4}\right)^2[/tex]
[tex]x^2-\frac{7x}{2}+\left(-\frac{7}{4}\right)^2=\frac{225}{16}[/tex]
[tex]\left(x-\frac{7}{4}\right)^2=\frac{225}{16}[/tex]
[tex]\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=\sqrt{a},\:-\sqrt{a}[/tex]
slove
[tex]x-\frac{7}{4}=\sqrt{\frac{225}{16}}[/tex]
[tex]x-\frac{7}{4}=\frac{\sqrt{225}}{\sqrt{16}}[/tex]
[tex]x-\frac{7}{4}=\frac{15}{4}[/tex]
[tex]\mathrm{Add\:}\frac{7}{4}\mathrm{\:to\:both\:sides}[/tex]
[tex]x-\frac{7}{4}+\frac{7}{4}=\frac{15}{4}+\frac{7}{4}[/tex]
[tex]x=\frac{11}{2}[/tex]
solving
[tex]x-\frac{7}{4}=-\sqrt{\frac{225}{16}}[/tex]
[tex]x-\frac{7}{4}=-\frac{15}{4}[/tex]
[tex]\mathrm{Add\:}\frac{7}{4}\mathrm{\:to\:both\:sides}[/tex]
[tex]x-\frac{7}{4}+\frac{7}{4}=-\frac{15}{4}+\frac{7}{4}[/tex]
[tex]x=-2[/tex]
The solutions to the quadratic equation are:
[tex]x=\frac{11}{2},\:x=-2[/tex]
Therefore, x=-2 solves the equation h(x)=10.
