Solve sqrt{7x-21}-7=2x-20

$$\sqrt { 7 x - 21 } - 7 = 2 x - 20$$

Solve for x

$$x=10$$

Solution Steps

Subtract -7 from both sides of the equation.

$$\sqrt{7x-21}=2x-20+7$$

Add -20 and 7 to get -13.

$$\sqrt{7x-21}=2x-13$$

Square both sides of the equation.

$$\left(\sqrt{7x-21}\right)^{2}=\left(2x-13\right)^{2}$$

Calculate \sqrt{7x-21} to the power of 2 and get 7x-21.

$$7x-21=\left(2x-13\right)^{2}$$

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

$$7x-21=4x^{2}-52x+169$$

Subtract 4x^{2} from both sides.

$$7x-21-4x^{2}=-52x+169$$

Add 52x to both sides.

$$7x-21-4x^{2}+52x=169$$

Combine 7x and 52x to get 59x.

$$59x-21-4x^{2}=169$$

Subtract 169 from both sides.

$$59x-21-4x^{2}-169=0$$

Subtract 169 from -21 to get -190.

$$59x-190-4x^{2}=0$$

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

$$-4x^{2}+59x-190=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-190. To find a and b, set up a system to be solved.

$$a+b=59 ab=-4\left(-190\right)=760$$

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 760.

$$1,760 2,380 4,190 5,152 8,95 10,76 19,40 20,38$$

Calculate the sum for each pair.

$$1+760=761 2+380=382 4+190=194 5+152=157 8+95=103 10+76=86 19+40=59 20+38=58$$

The solution is the pair that gives sum 59.

$$a=40 b=19$$

Rewrite -4x^{2}+59x-190 as \left(-4x^{2}+40x\right)+\left(19x-190\right).

$$\left(-4x^{2}+40x\right)+\left(19x-190\right)$$

Factor out 4x in the first and -19 in the second group.

$$4x\left(-x+10\right)-19\left(-x+10\right)$$

Factor out common term -x+10 by using distributive property.

$$\left(-x+10\right)\left(4x-19\right)$$

To find equation solutions, solve -x+10=0 and 4x-19=0.

$$x=10 x=\frac{19}{4}$$

Substitute 10 for x in the equation \sqrt{7x-21}-7=2x-20.

$$\sqrt{7\times 10-21}-7=2\times 10-20$$

Simplify. The value x=10 satisfies the equation.

$$0=0$$

Substitute \frac{19}{4} for x in the equation \sqrt{7x-21}-7=2x-20.

$$\sqrt{7\times \frac{19}{4}-21}-7=2\times \frac{19}{4}-20$$

Simplify. The value x=\frac{19}{4} does not satisfy the equation.

$$-\frac{7}{2}=-\frac{21}{2}$$

Equation \sqrt{7x-21}=2x-13 has a unique solution.

$$x=10$$