Solve x-6sqrt{x+1}+10=0

$$x-6 \sqrt{ x+1 } +10=0$$

Solve for x

$$x=8$$

Solution Steps

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$x-6\sqrt{x+1}=-10$$

Subtract x from both sides of the equation.

$$-6\sqrt{x+1}=-10-x$$

Square both sides of the equation.

$$\left(-6\sqrt{x+1}\right)^{2}=\left(-10-x\right)^{2}$$

Expand \left(-6\sqrt{x+1}\right)^{2}.

$$\left(-6\right)^{2}\left(\sqrt{x+1}\right)^{2}=\left(-10-x\right)^{2}$$

Calculate -6 to the power of 2 and get 36.

$$36\left(\sqrt{x+1}\right)^{2}=\left(-10-x\right)^{2}$$

Calculate \sqrt{x+1} to the power of 2 and get x+1.

$$36\left(x+1\right)=\left(-10-x\right)^{2}$$

Use the distributive property to multiply 36 by x+1.

$$36x+36=\left(-10-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-10-x\right)^{2}.

$$36x+36=100+20x+x^{2}$$

Subtract 20x from both sides.

$$36x+36-20x=100+x^{2}$$

Combine 36x and -20x to get 16x.

$$16x+36=100+x^{2}$$

Subtract x^{2} from both sides.

$$16x+36-x^{2}=100$$

Subtract 100 from both sides.

$$16x+36-x^{2}-100=0$$

Subtract 100 from 36 to get -64.

$$16x-64-x^{2}=0$$

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

$$-x^{2}+16x-64=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-64. To find a and b, set up a system to be solved.

$$a+b=16 ab=-\left(-64\right)=64$$

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 64.

$$1,64 2,32 4,16 8,8$$

Calculate the sum for each pair.

$$1+64=65 2+32=34 4+16=20 8+8=16$$

The solution is the pair that gives sum 16.

$$a=8 b=8$$

Rewrite -x^{2}+16x-64 as \left(-x^{2}+8x\right)+\left(8x-64\right).

$$\left(-x^{2}+8x\right)+\left(8x-64\right)$$

Factor out -x in the first and 8 in the second group.

$$-x\left(x-8\right)+8\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(-x+8\right)$$

To find equation solutions, solve x-8=0 and -x+8=0.

$$x=8 x=8$$

Substitute 8 for x in the equation x-6\sqrt{x+1}+10=0.

$$8-6\sqrt{8+1}+10=0$$

Simplify. The value x=8 satisfies the equation.

$$0=0$$

Substitute 8 for x in the equation x-6\sqrt{x+1}+10=0.

$$8-6\sqrt{8+1}+10=0$$

Simplify. The value x=8 satisfies the equation.

$$0=0$$

List all solutions of -6\sqrt{x+1}=-x-10.

$$x=8 x=8$$