Solve sqrt{3x+7}=2-2x

$$\sqrt { 3 x + 7 } = 2 - 2 x$$

Solve for x

$$x=-\frac{1}{4}=-0.25$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{3x+7}\right)^{2}=\left(2-2x\right)^{2}$$

Calculate \sqrt{3x+7} to the power of 2 and get 3x+7.

$$3x+7=\left(2-2x\right)^{2}$$

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

$$3x+7=4-8x+4x^{2}$$

Subtract 4 from both sides.

$$3x+7-4=-8x+4x^{2}$$

Subtract 4 from 7 to get 3.

$$3x+3=-8x+4x^{2}$$

Add 8x to both sides.

$$3x+3+8x=4x^{2}$$

Combine 3x and 8x to get 11x.

$$11x+3=4x^{2}$$

Subtract 4x^{2} from both sides.

$$11x+3-4x^{2}=0$$

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

$$-4x^{2}+11x+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=11 ab=-4\times 3=-12$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 11.

$$a=12 b=-1$$

Rewrite -4x^{2}+11x+3 as \left(-4x^{2}+12x\right)+\left(-x+3\right).

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

Factor out 4x in -4x^{2}+12x.

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

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(4x+1\right)$$

To find equation solutions, solve -x+3=0 and 4x+1=0.

$$x=3 x=-\frac{1}{4}$$

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

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

Simplify. The value x=3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$4=-4$$

Substitute -\frac{1}{4} for x in the equation \sqrt{3x+7}=2-2x.

$$\sqrt{3\left(-\frac{1}{4}\right)+7}=2-2\left(-\frac{1}{4}\right)$$

Simplify. The value x=-\frac{1}{4} satisfies the equation.

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

Equation \sqrt{3x+7}=2-2x has a unique solution.

$$x=-\frac{1}{4}$$