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

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

Solve for x

$$x=\frac{3}{4}=0.75$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{3x+10} to the power of 2 and get 3x+10.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-2x\right)^{2}.

$$3x+10=25-20x+4x^{2}$$

Subtract 25 from both sides.

$$3x+10-25=-20x+4x^{2}$$

Subtract 25 from 10 to get -15.

$$3x-15=-20x+4x^{2}$$

Add 20x to both sides.

$$3x-15+20x=4x^{2}$$

Combine 3x and 20x to get 23x.

$$23x-15=4x^{2}$$

Subtract 4x^{2} from both sides.

$$23x-15-4x^{2}=0$$

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

$$-4x^{2}+23x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=23 ab=-4\left(-15\right)=60$$

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 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Calculate the sum for each pair.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

The solution is the pair that gives sum 23.

$$a=20 b=3$$

Rewrite -4x^{2}+23x-15 as \left(-4x^{2}+20x\right)+\left(3x-15\right).

$$\left(-4x^{2}+20x\right)+\left(3x-15\right)$$

Factor out 4x in the first and -3 in the second group.

$$4x\left(-x+5\right)-3\left(-x+5\right)$$

Factor out common term -x+5 by using distributive property.

$$\left(-x+5\right)\left(4x-3\right)$$

To find equation solutions, solve -x+5=0 and 4x-3=0.

$$x=5 x=\frac{3}{4}$$

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

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

Simplify. The value x=5 does not satisfy the equation because the left and the right hand side have opposite signs.

$$5=-5$$

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

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

Simplify. The value x=\frac{3}{4} satisfies the equation.

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

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

$$x=\frac{3}{4}$$