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

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

Solve for x

$$x=-2$$

Solution Steps

Square both sides of the equation.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2x-3\right)^{2}.

$$4x^{2}+12x+9=\left(\sqrt{7+3x}\right)^{2}$$

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

$$4x^{2}+12x+9=7+3x$$

Subtract 7 from both sides.

$$4x^{2}+12x+9-7=3x$$

Subtract 7 from 9 to get 2.

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

Subtract 3x from both sides.

$$4x^{2}+12x+2-3x=0$$

Combine 12x and -3x to get 9x.

$$4x^{2}+9x+2=0$$

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

$$a+b=9 ab=4\times 2=8$$

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 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 9.

$$a=1 b=8$$

Rewrite 4x^{2}+9x+2 as \left(4x^{2}+x\right)+\left(8x+2\right).

$$\left(4x^{2}+x\right)+\left(8x+2\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(4x+1\right)+2\left(4x+1\right)$$

Factor out common term 4x+1 by using distributive property.

$$\left(4x+1\right)\left(x+2\right)$$

To find equation solutions, solve 4x+1=0 and x+2=0.

$$x=-\frac{1}{4} x=-2$$

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

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

Simplify. The value x=-\frac{1}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

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

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

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

Simplify. The value x=-2 satisfies the equation.

$$1=1$$

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

$$x=-2$$