Question

Solve:
√ (x²- 14 + 49) = x - 7

Answer 1

`\large\boxed{x\geq7\to x\in[7,\ \infty)}`

Explanation:

`√(x^2-14x+49)=x-7\\\\\underbrace{√(x^2-2(x)(7)+7^2)}_((*))=x-7\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\√((x-7)^2)=x-7\qquad\text{use}\ √(a^2)=|a|\\\\|x-7|=x-7\qquad\text{use the de}\text{finition of an absolute value}\\\\|x-7|=\left\{\begin{array}{ccc}x-7&for&x\geq7\\-(x-7)&for&x<7\end{array}\right`

`(1)\ x<7\to x\in(-\infty,\ 7)\\\\-(x-7)=x-7\\-x-(-7)=x-7\\-x+7=x-7\qquad\text{subtract 7 from both sides}\\-x=x-14\qquad\text{subtract x from both sides}\\-2x=-14\qquad\text{divide both sides by (-2)}\\x=7\\otin(-\infty,\ 7)\\\\NO\ SOLUTION`

`(2)\ x\geq7\to x\in[7,\ \infty)\\\\x-7=x-7\qquad\text{subtract x from both sides}\\-7=-7\qquad TRUE\\\\x\in[7,\ \infty)`

Answer 2

`Answer: \\ \sqrt{({x}^(2) - 14x + 49)} = x - 7 \\ \Leftrightarrow \sqrt{ {(x - 7)}^(2) } = x - 7 \\ \Leftrightarrow |x - 7| = x - 7 \\ \Leftrightarrow x - 7 = x - 7 \: \vee \: x - 7 = 7 - x \\ \Leftrightarrow x \in R \: \vee \:x = 7 \\ \Rightarrow x\in[7,\infty)`