Solve x+6sqrt{x}-16=0

$$x + 6 \sqrt { x } - 16 = 0$$

Solve for x

$$x=4$$

Solution Steps

Subtract x-16 from both sides of the equation.

$$6\sqrt{x}=-\left(x-16\right)$$

To find the opposite of x-16, find the opposite of each term.

$$6\sqrt{x}=-x-\left(-16\right)$$

The opposite of -16 is 16.

$$6\sqrt{x}=-x+16$$

Square both sides of the equation.

$$\left(6\sqrt{x}\right)^{2}=\left(-x+16\right)^{2}$$

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

$$6^{2}\left(\sqrt{x}\right)^{2}=\left(-x+16\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$36\left(\sqrt{x}\right)^{2}=\left(-x+16\right)^{2}$$

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

$$36x=\left(-x+16\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-x+16\right)^{2}.

$$36x=x^{2}-32x+256$$

Subtract x^{2} from both sides.

$$36x-x^{2}=-32x+256$$

Add 32x to both sides.

$$36x-x^{2}+32x=256$$

Combine 36x and 32x to get 68x.

$$68x-x^{2}=256$$

Subtract 256 from both sides.

$$68x-x^{2}-256=0$$

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

$$-x^{2}+68x-256=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-256. To find a and b, set up a system to be solved.

$$a+b=68 ab=-\left(-256\right)=256$$

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 256.

$$1,256 2,128 4,64 8,32 16,16$$

Calculate the sum for each pair.

$$1+256=257 2+128=130 4+64=68 8+32=40 16+16=32$$

The solution is the pair that gives sum 68.

$$a=64 b=4$$

Rewrite -x^{2}+68x-256 as \left(-x^{2}+64x\right)+\left(4x-256\right).

$$\left(-x^{2}+64x\right)+\left(4x-256\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(x-64\right)+4\left(x-64\right)$$

Factor out common term x-64 by using distributive property.

$$\left(x-64\right)\left(-x+4\right)$$

To find equation solutions, solve x-64=0 and -x+4=0.

$$x=64 x=4$$

Substitute 64 for x in the equation x+6\sqrt{x}-16=0.

$$64+6\sqrt{64}-16=0$$

Simplify. The value x=64 does not satisfy the equation.

$$96=0$$

Substitute 4 for x in the equation x+6\sqrt{x}-16=0.

$$4+6\sqrt{4}-16=0$$

Simplify. The value x=4 satisfies the equation.

$$0=0$$

Equation 6\sqrt{x}=16-x has a unique solution.

$$x=4$$