Solve sqrt{21-2x}=x-3

$$\sqrt { 21 - 2 x } = x - 3$$

Solve for x

$$x=6$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{21-2x}\right)^{2}=\left(x-3\right)^{2}$$

Calculate \sqrt{21-2x} to the power of 2 and get 21-2x.

$$21-2x=\left(x-3\right)^{2}$$

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

$$21-2x=x^{2}-6x+9$$

Subtract x^{2} from both sides.

$$21-2x-x^{2}=-6x+9$$

Add 6x to both sides.

$$21-2x-x^{2}+6x=9$$

Combine -2x and 6x to get 4x.

$$21+4x-x^{2}=9$$

Subtract 9 from both sides.

$$21+4x-x^{2}-9=0$$

Subtract 9 from 21 to get 12.

$$12+4x-x^{2}=0$$

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

$$-x^{2}+4x+12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=4 ab=-12=-12$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 4.

$$a=6 b=-2$$

Rewrite -x^{2}+4x+12 as \left(-x^{2}+6x\right)+\left(-2x+12\right).

$$\left(-x^{2}+6x\right)+\left(-2x+12\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(x-6\right)-2\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x-2\right)$$

To find equation solutions, solve x-6=0 and -x-2=0.

$$x=6 x=-2$$

Substitute 6 for x in the equation \sqrt{21-2x}=x-3.

$$\sqrt{21-2\times 6}=6-3$$

Simplify. The value x=6 satisfies the equation.

$$3=3$$

Substitute -2 for x in the equation \sqrt{21-2x}=x-3.

$$\sqrt{21-2\left(-2\right)}=-2-3$$

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$5=-5$$

Equation \sqrt{21-2x}=x-3 has a unique solution.

$$x=6$$