Answer:
The solutions to the quadratic equations will be:
[tex]x=\sqrt{57}+6,\:x=-\sqrt{57}+6[/tex]
Step-by-step explanation:
Given the expression
[tex]x^2-12x\:=\:21[/tex]
Let us solve the equation by completing the square
[tex]x^2-12x\:=\:21[/tex]
Add (-6)² to both sides
[tex]x^2-12x+\left(-6\right)^2=21+\left(-6\right)^2[/tex]
simplify
[tex]x^2-12x+\left(-6\right)^2=57[/tex]
Apply perfect square formula: (a-b)² = a²-2ab+b²
i.e.
[tex]x^2-12x+\left(-6\right)^2=\left(x-6\right)^2[/tex]
so the expression becomes
[tex]\left(x-6\right)^2=57[/tex]
[tex]\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=\sqrt{a},\:-\sqrt{a}[/tex]
solve
[tex]x-6=\sqrt{57}[/tex]
add 6 to both sides
[tex]x-6+6=\sqrt{57}+6[/tex]
Simplify
[tex]x=\sqrt{57}+6[/tex]
also solving
[tex]x-6=-\sqrt{57}[/tex]
add 6 to both sides
[tex]x-6+6=-\sqrt{57}+6[/tex]
Simplify
[tex]x=-\sqrt{57}+6[/tex]
Therefore, the solutions to the quadratic equation will be:
[tex]x=\sqrt{57}+6,\:x=-\sqrt{57}+6[/tex]
