Solve sqrt{3x-5}+10=2x

$$\sqrt { 3 x - 5 } + 10 = 2 x$$

Solve for x

$$x=7$$

Solution Steps

Subtract 10 from both sides of the equation.

$$\sqrt{3x-5}=2x-10$$

Square both sides of the equation.

$$\left(\sqrt{3x-5}\right)^{2}=\left(2x-10\right)^{2}$$

Calculate \sqrt{3x-5} to the power of 2 and get 3x-5.

$$3x-5=\left(2x-10\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-10\right)^{2}.

$$3x-5=4x^{2}-40x+100$$

Subtract 4x^{2} from both sides.

$$3x-5-4x^{2}=-40x+100$$

Add 40x to both sides.

$$3x-5-4x^{2}+40x=100$$

Combine 3x and 40x to get 43x.

$$43x-5-4x^{2}=100$$

Subtract 100 from both sides.

$$43x-5-4x^{2}-100=0$$

Subtract 100 from -5 to get -105.

$$43x-105-4x^{2}=0$$

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

$$-4x^{2}+43x-105=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-105. To find a and b, set up a system to be solved.

$$a+b=43 ab=-4\left(-105\right)=420$$

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 420.

$$1,420 2,210 3,140 4,105 5,84 6,70 7,60 10,42 12,35 14,30 15,28 20,21$$

Calculate the sum for each pair.

$$1+420=421 2+210=212 3+140=143 4+105=109 5+84=89 6+70=76 7+60=67 10+42=52 12+35=47 14+30=44 15+28=43 20+21=41$$

The solution is the pair that gives sum 43.

$$a=28 b=15$$

Rewrite -4x^{2}+43x-105 as \left(-4x^{2}+28x\right)+\left(15x-105\right).

$$\left(-4x^{2}+28x\right)+\left(15x-105\right)$$

Factor out 4x in the first and -15 in the second group.

$$4x\left(-x+7\right)-15\left(-x+7\right)$$

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

$$\left(-x+7\right)\left(4x-15\right)$$

To find equation solutions, solve -x+7=0 and 4x-15=0.

$$x=7 x=\frac{15}{4}$$

Substitute 7 for x in the equation \sqrt{3x-5}+10=2x.

$$\sqrt{3\times 7-5}+10=2\times 7$$

Simplify. The value x=7 satisfies the equation.

$$14=14$$

Substitute \frac{15}{4} for x in the equation \sqrt{3x-5}+10=2x.

$$\sqrt{3\times \frac{15}{4}-5}+10=2\times \frac{15}{4}$$

Simplify. The value x=\frac{15}{4} does not satisfy the equation.

$$\frac{25}{2}=\frac{15}{2}$$

Equation \sqrt{3x-5}=2x-10 has a unique solution.

$$x=7$$