Solve sqrt{3x+4}=4-2x

$$\sqrt { 3 x + 4 } = 4 - 2 x$$

Solve for x

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

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{3x+4}\right)^{2}=\left(4-2x\right)^{2}$$

Calculate \sqrt{3x+4} to the power of 2 and get 3x+4.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-2x\right)^{2}.

$$3x+4=16-16x+4x^{2}$$

Subtract 16 from both sides.

$$3x+4-16=-16x+4x^{2}$$

Subtract 16 from 4 to get -12.

$$3x-12=-16x+4x^{2}$$

Add 16x to both sides.

$$3x-12+16x=4x^{2}$$

Combine 3x and 16x to get 19x.

$$19x-12=4x^{2}$$

Subtract 4x^{2} from both sides.

$$19x-12-4x^{2}=0$$

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

$$-4x^{2}+19x-12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=19 ab=-4\left(-12\right)=48$$

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 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 19.

$$a=16 b=3$$

Rewrite -4x^{2}+19x-12 as \left(-4x^{2}+16x\right)+\left(3x-12\right).

$$\left(-4x^{2}+16x\right)+\left(3x-12\right)$$

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

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

Factor out common term -x+4 by using distributive property.

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

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

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

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

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

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

$$4=-4$$

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

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

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

$$\frac{5}{2}=\frac{5}{2}$$

Equation \sqrt{3x+4}=4-2x has a unique solution.

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