Solve 10-4sqrt{10+x}=x+8

$$10 - 4 \sqrt { 10 + x } = x + 8$$

Solve for x

$$x=-6$$

Solution Steps

Subtract 10 from both sides of the equation.

$$-4\sqrt{10+x}=x+8-10$$

Subtract 10 from 8 to get -2.

$$-4\sqrt{10+x}=x-2$$

Square both sides of the equation.

$$\left(-4\sqrt{10+x}\right)^{2}=\left(x-2\right)^{2}$$

Expand \left(-4\sqrt{10+x}\right)^{2}.

$$\left(-4\right)^{2}\left(\sqrt{10+x}\right)^{2}=\left(x-2\right)^{2}$$

Calculate -4 to the power of 2 and get 16.

$$16\left(\sqrt{10+x}\right)^{2}=\left(x-2\right)^{2}$$

Calculate \sqrt{10+x} to the power of 2 and get 10+x.

$$16\left(10+x\right)=\left(x-2\right)^{2}$$

Use the distributive property to multiply 16 by 10+x.

$$160+16x=\left(x-2\right)^{2}$$

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

$$160+16x=x^{2}-4x+4$$

Subtract x^{2} from both sides.

$$160+16x-x^{2}=-4x+4$$

Add 4x to both sides.

$$160+16x-x^{2}+4x=4$$

Combine 16x and 4x to get 20x.

$$160+20x-x^{2}=4$$

Subtract 4 from both sides.

$$160+20x-x^{2}-4=0$$

Subtract 4 from 160 to get 156.

$$156+20x-x^{2}=0$$

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

$$-x^{2}+20x+156=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+156. To find a and b, set up a system to be solved.

$$a+b=20 ab=-156=-156$$

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 -156.

$$-1,156 -2,78 -3,52 -4,39 -6,26 -12,13$$

Calculate the sum for each pair.

$$-1+156=155 -2+78=76 -3+52=49 -4+39=35 -6+26=20 -12+13=1$$

The solution is the pair that gives sum 20.

$$a=26 b=-6$$

Rewrite -x^{2}+20x+156 as \left(-x^{2}+26x\right)+\left(-6x+156\right).

$$\left(-x^{2}+26x\right)+\left(-6x+156\right)$$

Factor out -x in the first and -6 in the second group.

$$-x\left(x-26\right)-6\left(x-26\right)$$

Factor out common term x-26 by using distributive property.

$$\left(x-26\right)\left(-x-6\right)$$

To find equation solutions, solve x-26=0 and -x-6=0.

$$x=26 x=-6$$

Substitute 26 for x in the equation 10-4\sqrt{10+x}=x+8.

$$10-4\sqrt{10+26}=26+8$$

Simplify. The value x=26 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-14=34$$

Substitute -6 for x in the equation 10-4\sqrt{10+x}=x+8.

$$10-4\sqrt{10-6}=-6+8$$

Simplify. The value x=-6 satisfies the equation.

$$2=2$$

Equation -4\sqrt{x+10}=x-2 has a unique solution.

$$x=-6$$