Solve sqrt{-3x+28}=x

$$\sqrt { - 3 x + 28 } = x$$

Solve for x

$$x=4$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{-3x+28}\right)^{2}=x^{2}$$

Calculate \sqrt{-3x+28} to the power of 2 and get -3x+28.

$$-3x+28=x^{2}$$

Subtract x^{2} from both sides.

$$-3x+28-x^{2}=0$$

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

$$-x^{2}-3x+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-28=-28$$

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 -28.

$$1,-28 2,-14 4,-7$$

Calculate the sum for each pair.

$$1-28=-27 2-14=-12 4-7=-3$$

The solution is the pair that gives sum -3.

$$a=4 b=-7$$

Rewrite -x^{2}-3x+28 as \left(-x^{2}+4x\right)+\left(-7x+28\right).

$$\left(-x^{2}+4x\right)+\left(-7x+28\right)$$

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

$$x\left(-x+4\right)+7\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(x+7\right)$$

To find equation solutions, solve -x+4=0 and x+7=0.

$$x=4 x=-7$$

Substitute 4 for x in the equation \sqrt{-3x+28}=x.

$$\sqrt{-3\times 4+28}=4$$

Simplify. The value x=4 satisfies the equation.

$$4=4$$

Substitute -7 for x in the equation \sqrt{-3x+28}=x.

$$\sqrt{-3\left(-7\right)+28}=-7$$

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{28-3x}=x has a unique solution.

$$x=4$$