Solve sqrt{7x-38}=2x-13

$$\sqrt{ 7x-38 } = 2x-13$$

Solve for x

$$x=9$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{7x-38}\right)^{2}=\left(2x-13\right)^{2}$$

Calculate \sqrt{7x-38} to the power of 2 and get 7x-38.

$$7x-38=\left(2x-13\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-13\right)^{2}.

$$7x-38=4x^{2}-52x+169$$

Subtract 4x^{2} from both sides.

$$7x-38-4x^{2}=-52x+169$$

Add 52x to both sides.

$$7x-38-4x^{2}+52x=169$$

Combine 7x and 52x to get 59x.

$$59x-38-4x^{2}=169$$

Subtract 169 from both sides.

$$59x-38-4x^{2}-169=0$$

Subtract 169 from -38 to get -207.

$$59x-207-4x^{2}=0$$

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

$$-4x^{2}+59x-207=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-207. To find a and b, set up a system to be solved.

$$a+b=59 ab=-4\left(-207\right)=828$$

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 828.

$$1,828 2,414 3,276 4,207 6,138 9,92 12,69 18,46 23,36$$

Calculate the sum for each pair.

$$1+828=829 2+414=416 3+276=279 4+207=211 6+138=144 9+92=101 12+69=81 18+46=64 23+36=59$$

The solution is the pair that gives sum 59.

$$a=36 b=23$$

Rewrite -4x^{2}+59x-207 as \left(-4x^{2}+36x\right)+\left(23x-207\right).

$$\left(-4x^{2}+36x\right)+\left(23x-207\right)$$

Factor out 4x in the first and -23 in the second group.

$$4x\left(-x+9\right)-23\left(-x+9\right)$$

Factor out common term -x+9 by using distributive property.

$$\left(-x+9\right)\left(4x-23\right)$$

To find equation solutions, solve -x+9=0 and 4x-23=0.

$$x=9 x=\frac{23}{4}$$

Substitute 9 for x in the equation \sqrt{7x-38}=2x-13.

$$\sqrt{7\times 9-38}=2\times 9-13$$

Simplify. The value x=9 satisfies the equation.

$$5=5$$

Substitute \frac{23}{4} for x in the equation \sqrt{7x-38}=2x-13.

$$\sqrt{7\times \frac{23}{4}-38}=2\times \frac{23}{4}-13$$

Simplify. The value x=\frac{23}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{3}{2}=-\frac{3}{2}$$

Equation \sqrt{7x-38}=2x-13 has a unique solution.

$$x=9$$