Solve 3sqrt{x+5}-5=x

$$3 \sqrt { x + 5 } - 5 = x$$

Solve for x

$$x=4$$

$$x=-5$$

Solution Steps

Subtract -5 from both sides of the equation.

$$3\sqrt{x+5}=x+5$$

Square both sides of the equation.

$$\left(3\sqrt{x+5}\right)^{2}=\left(x+5\right)^{2}$$

Expand \left(3\sqrt{x+5}\right)^{2}.

$$3^{2}\left(\sqrt{x+5}\right)^{2}=\left(x+5\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{x+5}\right)^{2}=\left(x+5\right)^{2}$$

Calculate \sqrt{x+5} to the power of 2 and get x+5.

$$9\left(x+5\right)=\left(x+5\right)^{2}$$

Use the distributive property to multiply 9 by x+5.

$$9x+45=\left(x+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$9x+45=x^{2}+10x+25$$

Subtract x^{2} from both sides.

$$9x+45-x^{2}=10x+25$$

Subtract 10x from both sides.

$$9x+45-x^{2}-10x=25$$

Combine 9x and -10x to get -x.

$$-x+45-x^{2}=25$$

Subtract 25 from both sides.

$$-x+45-x^{2}-25=0$$

Subtract 25 from 45 to get 20.

$$-x+20-x^{2}=0$$

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

$$-x^{2}-x+20=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+20. To find a and b, set up a system to be solved.

$$a+b=-1 ab=-20=-20$$

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 -20.

$$1,-20 2,-10 4,-5$$

Calculate the sum for each pair.

$$1-20=-19 2-10=-8 4-5=-1$$

The solution is the pair that gives sum -1.

$$a=4 b=-5$$

Rewrite -x^{2}-x+20 as \left(-x^{2}+4x\right)+\left(-5x+20\right).

$$\left(-x^{2}+4x\right)+\left(-5x+20\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(-x+4\right)+5\left(-x+4\right)$$

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

$$\left(-x+4\right)\left(x+5\right)$$

To find equation solutions, solve -x+4=0 and x+5=0.

$$x=4 x=-5$$

Substitute 4 for x in the equation 3\sqrt{x+5}-5=x.

$$3\sqrt{4+5}-5=4$$

Simplify. The value x=4 satisfies the equation.

$$4=4$$

Substitute -5 for x in the equation 3\sqrt{x+5}-5=x.

$$3\sqrt{-5+5}-5=-5$$

Simplify. The value x=-5 satisfies the equation.

$$-5=-5$$

List all solutions of 3\sqrt{x+5}=x+5.

$$x=4 x=-5$$