Solve sqrt{21-4x}=x

$$\sqrt { 21 - 4 x } = x$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{21-4x}\right)^{2}=x^{2}$$

Calculate \sqrt{21-4x} to the power of 2 and get 21-4x.

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

Subtract x^{2} from both sides.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-21=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -21.

$$1,-21 3,-7$$

Calculate the sum for each pair.

$$1-21=-20 3-7=-4$$

The solution is the pair that gives sum -4.

$$a=3 b=-7$$

Rewrite -x^{2}-4x+21 as \left(-x^{2}+3x\right)+\left(-7x+21\right).

$$\left(-x^{2}+3x\right)+\left(-7x+21\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(-x+3\right)+7\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(x+7\right)$$

To find equation solutions, solve -x+3=0 and x+7=0.

$$x=3 x=-7$$

Substitute 3 for x in the equation \sqrt{21-4x}=x.

$$\sqrt{21-4\times 3}=3$$

Simplify. The value x=3 satisfies the equation.

$$3=3$$

Substitute -7 for x in the equation \sqrt{21-4x}=x.

$$\sqrt{21-4\left(-7\right)}=-7$$

Simplify. The value x=-7 does not satisfy the equation because the left and the right hand side have opposite signs.

$$7=-7$$

Equation \sqrt{21-4x}=x has a unique solution.

$$x=3$$