Solve 4sqrt{x-2}=x+1

$$4 \sqrt { x - 2 } = x + 1$$

Solve for x

$$x=11$$

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(4\sqrt{x-2}\right)^{2}=\left(x+1\right)^{2}$$

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

$$4^{2}\left(\sqrt{x-2}\right)^{2}=\left(x+1\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16\left(\sqrt{x-2}\right)^{2}=\left(x+1\right)^{2}$$

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

$$16\left(x-2\right)=\left(x+1\right)^{2}$$

Use the distributive property to multiply 16 by x-2.

$$16x-32=\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$16x-32=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$16x-32-x^{2}=2x+1$$

Subtract 2x from both sides.

$$16x-32-x^{2}-2x=1$$

Combine 16x and -2x to get 14x.

$$14x-32-x^{2}=1$$

Subtract 1 from both sides.

$$14x-32-x^{2}-1=0$$

Subtract 1 from -32 to get -33.

$$14x-33-x^{2}=0$$

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

$$-x^{2}+14x-33=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-33. To find a and b, set up a system to be solved.

$$a+b=14 ab=-\left(-33\right)=33$$

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 33.

$$1,33 3,11$$

Calculate the sum for each pair.

$$1+33=34 3+11=14$$

The solution is the pair that gives sum 14.

$$a=11 b=3$$

Rewrite -x^{2}+14x-33 as \left(-x^{2}+11x\right)+\left(3x-33\right).

$$\left(-x^{2}+11x\right)+\left(3x-33\right)$$

Factor out -x in the first and 3 in the second group.

$$-x\left(x-11\right)+3\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(-x+3\right)$$

To find equation solutions, solve x-11=0 and -x+3=0.

$$x=11 x=3$$

Substitute 11 for x in the equation 4\sqrt{x-2}=x+1.

$$4\sqrt{11-2}=11+1$$

Simplify. The value x=11 satisfies the equation.

$$12=12$$

Substitute 3 for x in the equation 4\sqrt{x-2}=x+1.

$$4\sqrt{3-2}=3+1$$

Simplify. The value x=3 satisfies the equation.

$$4=4$$

List all solutions of 4\sqrt{x-2}=x+1.

$$x=11 x=3$$