Answer:
x =0 ,x =7
Step-by-step explanation:
Method 1 ; Factoring
[tex]9x^2 -63x =0\\\\\mathrm{Solve\:by\:factoring}\\\mathrm{Factor\:}9x^2-63x:\quad 9x\left(x-7\right)\\\\9x\left(x-7\right)=0\\\\\mathrm{Using\:the\:Zero\:Factor\:Principle:}\\if\:ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0\:\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)\\\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)\\\\x =0\\\mathrm{Solve\:}\:x-7=0:\quad x=7\\x=0,\:x=7[/tex]
Method 2 ; Completing the square
[tex]9x^2-63x=0\\\\\mathrm{Divide\:both\:sides\:by\:}9\\\frac{9x^2-63x}{9}=\frac{0}{9}\\\\x^2-7x=0\\\mathrm{Solve\:by\:completing\:the\:square}\\\mathrm{Write\:equation\:in\:the\:form:\:\:}\\x^2+2ax+a^2=\left(x+a\right)^2\\\\\mathrm{Solve\:for\:}a,\:2ax=-7x:\quad a=-\frac{7}{2}\\\\\mathrm{Add\:}a^2=\left(-\frac{7}{2}\right)^2\mathrm{\:to\:both\:sides}\\x^2-7x+\left(-\frac{7}{2}\right)^2=0+\left(-\frac{7}{2}\right)^2\\\\\mathrm{Simplify\:}\\0+\left(-\frac{7}{2}\right)^2:\quad \frac{49}{4}[/tex]
[tex]x^2-7x+\left(-\frac{7}{2}\right)^2=\frac{49}{4}\\\\\mathrm{Complete\:the\:square}\\\left(x-\frac{7}{2}\right)^2=\frac{49}{4}\\\\\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=\sqrt{a},\:-\sqrt{a}\\\\\mathrm{Solve\:}\:x-\frac{7}{2}=\sqrt{\frac{49}{4}}:\quad x=7\\\\\mathrm{Solve\:}\:x-\frac{7}{2}=-\sqrt{\frac{49}{4}}:\quad x=0\\\\x=7,\:x=0[/tex]
