Solve sqrt{3m-2}+m=10

$$\sqrt { 3 m - 2 } + m = 10$$

Solve for m

$$m=6$$

Solution Steps

Subtract m from both sides of the equation.

$$\sqrt{3m-2}=10-m$$

Square both sides of the equation.

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

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

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

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

$$3m-2=100-20m+m^{2}$$

Subtract 100 from both sides.

$$3m-2-100=-20m+m^{2}$$

Subtract 100 from -2 to get -102.

$$3m-102=-20m+m^{2}$$

Add 20m to both sides.

$$3m-102+20m=m^{2}$$

Combine 3m and 20m to get 23m.

$$23m-102=m^{2}$$

Subtract m^{2} from both sides.

$$23m-102-m^{2}=0$$

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

$$-m^{2}+23m-102=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -m^{2}+am+bm-102. To find a and b, set up a system to be solved.

$$a+b=23 ab=-\left(-102\right)=102$$

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 102.

$$1,102 2,51 3,34 6,17$$

Calculate the sum for each pair.

$$1+102=103 2+51=53 3+34=37 6+17=23$$

The solution is the pair that gives sum 23.

$$a=17 b=6$$

Rewrite -m^{2}+23m-102 as \left(-m^{2}+17m\right)+\left(6m-102\right).

$$\left(-m^{2}+17m\right)+\left(6m-102\right)$$

Factor out -m in the first and 6 in the second group.

$$-m\left(m-17\right)+6\left(m-17\right)$$

Factor out common term m-17 by using distributive property.

$$\left(m-17\right)\left(-m+6\right)$$

To find equation solutions, solve m-17=0 and -m+6=0.

$$m=17 m=6$$

Substitute 17 for m in the equation \sqrt{3m-2}+m=10.

$$\sqrt{3\times 17-2}+17=10$$

Simplify. The value m=17 does not satisfy the equation.

$$24=10$$

Substitute 6 for m in the equation \sqrt{3m-2}+m=10.

$$\sqrt{3\times 6-2}+6=10$$

Simplify. The value m=6 satisfies the equation.

$$10=10$$

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

$$m=6$$