Solve 4*sqrt{x}=12-x

$$4 \cdot \sqrt { x } = 12 - x$$

Solve for x

$$x=4$$

Solution Steps

Square both sides of the equation.

$$\left(4\sqrt{x}\right)^{2}=\left(12-x\right)^{2}$$

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

$$4^{2}\left(\sqrt{x}\right)^{2}=\left(12-x\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16\left(\sqrt{x}\right)^{2}=\left(12-x\right)^{2}$$

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

$$16x=\left(12-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-x\right)^{2}.

$$16x=144-24x+x^{2}$$

Subtract 144 from both sides.

$$16x-144=-24x+x^{2}$$

Add 24x to both sides.

$$16x-144+24x=x^{2}$$

Combine 16x and 24x to get 40x.

$$40x-144=x^{2}$$

Subtract x^{2} from both sides.

$$40x-144-x^{2}=0$$

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

$$-x^{2}+40x-144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-144. To find a and b, set up a system to be solved.

$$a+b=40 ab=-\left(-144\right)=144$$

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 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Calculate the sum for each pair.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

The solution is the pair that gives sum 40.

$$a=36 b=4$$

Rewrite -x^{2}+40x-144 as \left(-x^{2}+36x\right)+\left(4x-144\right).

$$\left(-x^{2}+36x\right)+\left(4x-144\right)$$

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

$$-x\left(x-36\right)+4\left(x-36\right)$$

Factor out common term x-36 by using distributive property.

$$\left(x-36\right)\left(-x+4\right)$$

To find equation solutions, solve x-36=0 and -x+4=0.

$$x=36 x=4$$

Substitute 36 for x in the equation 4\sqrt{x}=12-x.

$$4\sqrt{36}=12-36$$

Simplify. The value x=36 does not satisfy the equation because the left and the right hand side have opposite signs.

$$24=-24$$

Substitute 4 for x in the equation 4\sqrt{x}=12-x.

$$4\sqrt{4}=12-4$$

Simplify. The value x=4 satisfies the equation.

$$8=8$$

Equation 4\sqrt{x}=12-x has a unique solution.

$$x=4$$