Solve sqrt{6x+7}-(2x-7)=0

$$\sqrt{ 6x+7 } -(2x-7)=0$$

Solve for x

$$x=7$$

Solution Steps

Subtract -\left(2x-7\right) from both sides of the equation.

$$\sqrt{6x+7}=2x-7$$

Square both sides of the equation.

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

Calculate \sqrt{6x+7} to the power of 2 and get 6x+7.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-7\right)^{2}.

$$6x+7=4x^{2}-28x+49$$

Subtract 4x^{2} from both sides.

$$6x+7-4x^{2}=-28x+49$$

Add 28x to both sides.

$$6x+7-4x^{2}+28x=49$$

Combine 6x and 28x to get 34x.

$$34x+7-4x^{2}=49$$

Subtract 49 from both sides.

$$34x+7-4x^{2}-49=0$$

Subtract 49 from 7 to get -42.

$$34x-42-4x^{2}=0$$

Divide both sides by 2.

$$17x-21-2x^{2}=0$$

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

$$-2x^{2}+17x-21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=17 ab=-2\left(-21\right)=42$$

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 42.

$$1,42 2,21 3,14 6,7$$

Calculate the sum for each pair.

$$1+42=43 2+21=23 3+14=17 6+7=13$$

The solution is the pair that gives sum 17.

$$a=14 b=3$$

Rewrite -2x^{2}+17x-21 as \left(-2x^{2}+14x\right)+\left(3x-21\right).

$$\left(-2x^{2}+14x\right)+\left(3x-21\right)$$

Factor out 2x in the first and -3 in the second group.

$$2x\left(-x+7\right)-3\left(-x+7\right)$$

Factor out common term -x+7 by using distributive property.

$$\left(-x+7\right)\left(2x-3\right)$$

To find equation solutions, solve -x+7=0 and 2x-3=0.

$$x=7 x=\frac{3}{2}$$

Substitute 7 for x in the equation \sqrt{6x+7}-\left(2x-7\right)=0.

$$\sqrt{6\times 7+7}-\left(2\times 7-7\right)=0$$

Simplify. The value x=7 satisfies the equation.

$$0=0$$

Substitute \frac{3}{2} for x in the equation \sqrt{6x+7}-\left(2x-7\right)=0.

$$\sqrt{6\times \frac{3}{2}+7}-\left(2\times \frac{3}{2}-7\right)=0$$

Simplify. The value x=\frac{3}{2} does not satisfy the equation.

$$8=0$$

Equation \sqrt{6x+7}=2x-7 has a unique solution.

$$x=7$$