Solve sqrt{x+3}=4x+7

$$\sqrt { x + 3 } = 4 x + 7$$

Solve for x

$$x = -\frac{23}{16} = -1\frac{7}{16} = -1.4375$$

Solution Steps

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+7\right)^{2}.

$$x+3=16x^{2}+56x+49$$

Subtract 16x^{2} from both sides.

$$x+3-16x^{2}=56x+49$$

Subtract 56x from both sides.

$$x+3-16x^{2}-56x=49$$

Combine x and -56x to get -55x.

$$-55x+3-16x^{2}=49$$

Subtract 49 from both sides.

$$-55x+3-16x^{2}-49=0$$

Subtract 49 from 3 to get -46.

$$-55x-46-16x^{2}=0$$

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

$$-16x^{2}-55x-46=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -16x^{2}+ax+bx-46. To find a and b, set up a system to be solved.

$$a+b=-55 ab=-16\left(-46\right)=736$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 736.

$$-1,-736 -2,-368 -4,-184 -8,-92 -16,-46 -23,-32$$

Calculate the sum for each pair.

$$-1-736=-737 -2-368=-370 -4-184=-188 -8-92=-100 -16-46=-62 -23-32=-55$$

The solution is the pair that gives sum -55.

$$a=-23 b=-32$$

Rewrite -16x^{2}-55x-46 as \left(-16x^{2}-23x\right)+\left(-32x-46\right).

$$\left(-16x^{2}-23x\right)+\left(-32x-46\right)$$

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

$$-x\left(16x+23\right)-2\left(16x+23\right)$$

Factor out common term 16x+23 by using distributive property.

$$\left(16x+23\right)\left(-x-2\right)$$

To find equation solutions, solve 16x+23=0 and -x-2=0.

$$x=-\frac{23}{16} x=-2$$

Substitute -\frac{23}{16} for x in the equation \sqrt{x+3}=4x+7.

$$\sqrt{-\frac{23}{16}+3}=4\left(-\frac{23}{16}\right)+7$$

Simplify. The value x=-\frac{23}{16} satisfies the equation.

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

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

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

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Equation \sqrt{x+3}=4x+7 has a unique solution.

$$x=-\frac{23}{16}$$