Solve sqrt{x+2}-2sqrt{x+7}=-4

$$\sqrt { x + 2 } - 2 \sqrt { x + 7 } = - 4$$

Solve for x

$$x=2$$

$$x = -\frac{14}{9} = -1\frac{5}{9} \approx -1.555555556$$

Solution Steps

Subtract -2\sqrt{x+7} from both sides of the equation.

$$\sqrt{x+2}=-4+2\sqrt{x+7}$$

Square both sides of the equation.

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

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

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

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

$$x+2=16-16\sqrt{x+7}+4\left(\sqrt{x+7}\right)^{2}$$

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

$$x+2=16-16\sqrt{x+7}+4\left(x+7\right)$$

Use the distributive property to multiply 4 by x+7.

$$x+2=16-16\sqrt{x+7}+4x+28$$

Add 16 and 28 to get 44.

$$x+2=44-16\sqrt{x+7}+4x$$

Subtract 44+4x from both sides of the equation.

$$x+2-\left(44+4x\right)=-16\sqrt{x+7}$$

To find the opposite of 44+4x, find the opposite of each term.

$$x+2-44-4x=-16\sqrt{x+7}$$

Subtract 44 from 2 to get -42.

$$x-42-4x=-16\sqrt{x+7}$$

Combine x and -4x to get -3x.

$$-3x-42=-16\sqrt{x+7}$$

Square both sides of the equation.

$$\left(-3x-42\right)^{2}=\left(-16\sqrt{x+7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-3x-42\right)^{2}.

$$9x^{2}+252x+1764=\left(-16\sqrt{x+7}\right)^{2}$$

Expand \left(-16\sqrt{x+7}\right)^{2}.

$$9x^{2}+252x+1764=\left(-16\right)^{2}\left(\sqrt{x+7}\right)^{2}$$

Calculate -16 to the power of 2 and get 256.

$$9x^{2}+252x+1764=256\left(\sqrt{x+7}\right)^{2}$$

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

$$9x^{2}+252x+1764=256\left(x+7\right)$$

Use the distributive property to multiply 256 by x+7.

$$9x^{2}+252x+1764=256x+1792$$

Subtract 256x from both sides.

$$9x^{2}+252x+1764-256x=1792$$

Combine 252x and -256x to get -4x.

$$9x^{2}-4x+1764=1792$$

Subtract 1792 from both sides.

$$9x^{2}-4x+1764-1792=0$$

Subtract 1792 from 1764 to get -28.

$$9x^{2}-4x-28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx-28. To find a and b, set up a system to be solved.

$$a+b=-4 ab=9\left(-28\right)=-252$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -252.

$$1,-252 2,-126 3,-84 4,-63 6,-42 7,-36 9,-28 12,-21 14,-18$$

Calculate the sum for each pair.

$$1-252=-251 2-126=-124 3-84=-81 4-63=-59 6-42=-36 7-36=-29 9-28=-19 12-21=-9 14-18=-4$$

The solution is the pair that gives sum -4.

$$a=-18 b=14$$

Rewrite 9x^{2}-4x-28 as \left(9x^{2}-18x\right)+\left(14x-28\right).

$$\left(9x^{2}-18x\right)+\left(14x-28\right)$$

Factor out 9x in the first and 14 in the second group.

$$9x\left(x-2\right)+14\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(9x+14\right)$$

To find equation solutions, solve x-2=0 and 9x+14=0.

$$x=2 x=-\frac{14}{9}$$

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

$$\sqrt{2+2}-2\sqrt{2+7}=-4$$

Simplify. The value x=2 satisfies the equation.

$$-4=-4$$

Substitute -\frac{14}{9} for x in the equation \sqrt{x+2}-2\sqrt{x+7}=-4.

$$\sqrt{-\frac{14}{9}+2}-2\sqrt{-\frac{14}{9}+7}=-4$$

Simplify. The value x=-\frac{14}{9} satisfies the equation.

$$-4=-4$$

List all solutions of \sqrt{x+2}=2\sqrt{x+7}-4.

$$x=2 x=-\frac{14}{9}$$