Question

What is the positive solution to the equation x² - 6x + 9 = 16?

Answer 1

Using Factorization (method 1)

So we now rewrite -6x as the sum of the two factors -7x + 1x

so that we have

`\begin{gathered} x^2-7x+1x-7\text{ = }0 \\ We\text{ now group them} \\ (x^2-7x)(1x-7)=0 \\ \text{Then we factorize the brackets} \\ x(x-7)\text{ +1(x-7)=0} \\ so\text{ we pick the HCF/GCF, put them in a bracket and then pick one of the common brackets } \\ (x+1)(x-7)=0 \\ x\text{ +1 = 0 ; x = -1} \\ or \\ x-7\text{ = 0 ; x=7} \\ So\text{ the positive solution of the equation is x = 7} \end{gathered}`

Method 2

If we use the Quadratic formula

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

We must refer to the general quadratic equation

`ax^2+bx+c\text{ = 0}`
`\begin{gathered} \text{Comparing both equations} \\ a\text{ = 1} \\ b\text{ = -6} \\ c\text{ = -7} \end{gathered}`

Substituting these into the formula

`\begin{gathered} x=\frac{-(-6)\pm\sqrt[]{(-6)^2-4*1*-7}}{2\text{ x1}} \\ x\text{ = }\frac{6\pm\sqrt[]{36+28}}{2} \\ x=\text{ }\frac{6\pm\sqrt[]{64}}{2} \\ x=(6\pm8)/(2) \\ x=(6+8)/(2)=(14)/(2)=7 \\ or \\ x=(6-8)/(2)=(-2)/(2)=-1 \\ So\text{ the only positive solution of the equation is x=7} \end{gathered}`