Solve sqrt{7x-34}=x-6

$$\sqrt { 7 x - 34 } = x - 6$$

Solve for x

$$x=14$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{7x-34}\right)^{2}=\left(x-6\right)^{2}$$

Calculate \sqrt{7x-34} to the power of 2 and get 7x-34.

$$7x-34=\left(x-6\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$7x-34=x^{2}-12x+36$$

Subtract x^{2} from both sides.

$$7x-34-x^{2}=-12x+36$$

Add 12x to both sides.

$$7x-34-x^{2}+12x=36$$

Combine 7x and 12x to get 19x.

$$19x-34-x^{2}=36$$

Subtract 36 from both sides.

$$19x-34-x^{2}-36=0$$

Subtract 36 from -34 to get -70.

$$19x-70-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+19x-70=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-70. To find a and b, set up a system to be solved.

$$a+b=19 ab=-\left(-70\right)=70$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 70.

$$1,70 2,35 5,14 7,10$$

Calculate the sum for each pair.

$$1+70=71 2+35=37 5+14=19 7+10=17$$

The solution is the pair that gives sum 19.

$$a=14 b=5$$

Rewrite -x^{2}+19x-70 as \left(-x^{2}+14x\right)+\left(5x-70\right).

$$\left(-x^{2}+14x\right)+\left(5x-70\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-14\right)+5\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(-x+5\right)$$

To find equation solutions, solve x-14=0 and -x+5=0.

$$x=14 x=5$$

Substitute 14 for x in the equation \sqrt{7x-34}=x-6.

$$\sqrt{7\times 14-34}=14-6$$

Simplify. The value x=14 satisfies the equation.

$$8=8$$

Substitute 5 for x in the equation \sqrt{7x-34}=x-6.

$$\sqrt{7\times 5-34}=5-6$$

Simplify. The value x=5 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Equation \sqrt{7x-34}=x-6 has a unique solution.

$$x=14$$