Solve x-4sqrt{x-3}=3

$$x - 4 \sqrt { x - 3 } = 3$$

Solve for x

$$x=19$$

$$x=3$$

Solution Steps

Subtract x from both sides of the equation.

$$-4\sqrt{x-3}=3-x$$

Square both sides of the equation.

$$\left(-4\sqrt{x-3}\right)^{2}=\left(3-x\right)^{2}$$

Expand \left(-4\sqrt{x-3}\right)^{2}.

$$\left(-4\right)^{2}\left(\sqrt{x-3}\right)^{2}=\left(3-x\right)^{2}$$

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

$$16\left(\sqrt{x-3}\right)^{2}=\left(3-x\right)^{2}$$

Calculate \sqrt{x-3} to the power of 2 and get x-3.

$$16\left(x-3\right)=\left(3-x\right)^{2}$$

Use the distributive property to multiply 16 by x-3.

$$16x-48=\left(3-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-x\right)^{2}.

$$16x-48=9-6x+x^{2}$$

Subtract 9 from both sides.

$$16x-48-9=-6x+x^{2}$$

Subtract 9 from -48 to get -57.

$$16x-57=-6x+x^{2}$$

Add 6x to both sides.

$$16x-57+6x=x^{2}$$

Combine 16x and 6x to get 22x.

$$22x-57=x^{2}$$

Subtract x^{2} from both sides.

$$22x-57-x^{2}=0$$

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

$$-x^{2}+22x-57=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-57. To find a and b, set up a system to be solved.

$$a+b=22 ab=-\left(-57\right)=57$$

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 57.

$$1,57 3,19$$

Calculate the sum for each pair.

$$1+57=58 3+19=22$$

The solution is the pair that gives sum 22.

$$a=19 b=3$$

Rewrite -x^{2}+22x-57 as \left(-x^{2}+19x\right)+\left(3x-57\right).

$$\left(-x^{2}+19x\right)+\left(3x-57\right)$$

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

$$-x\left(x-19\right)+3\left(x-19\right)$$

Factor out common term x-19 by using distributive property.

$$\left(x-19\right)\left(-x+3\right)$$

To find equation solutions, solve x-19=0 and -x+3=0.

$$x=19 x=3$$

Substitute 19 for x in the equation x-4\sqrt{x-3}=3.

$$19-4\sqrt{19-3}=3$$

Simplify. The value x=19 satisfies the equation.

$$3=3$$

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

$$3-4\sqrt{3-3}=3$$

Simplify. The value x=3 satisfies the equation.

$$3=3$$

List all solutions of -4\sqrt{x-3}=3-x.

$$x=19 x=3$$