Solve 3sqrt{x+10}=x+6

$$3 \sqrt { x + 10 } = x + 6$$

Solve for x

$$x=6$$

Solution Steps

Square both sides of the equation.

$$\left(3\sqrt{x+10}\right)^{2}=\left(x+6\right)^{2}$$

Expand \left(3\sqrt{x+10}\right)^{2}.

$$3^{2}\left(\sqrt{x+10}\right)^{2}=\left(x+6\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{x+10}\right)^{2}=\left(x+6\right)^{2}$$

Calculate \sqrt{x+10} to the power of 2 and get x+10.

$$9\left(x+10\right)=\left(x+6\right)^{2}$$

Use the distributive property to multiply 9 by x+10.

$$9x+90=\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$9x+90=x^{2}+12x+36$$

Subtract x^{2} from both sides.

$$9x+90-x^{2}=12x+36$$

Subtract 12x from both sides.

$$9x+90-x^{2}-12x=36$$

Combine 9x and -12x to get -3x.

$$-3x+90-x^{2}=36$$

Subtract 36 from both sides.

$$-3x+90-x^{2}-36=0$$

Subtract 36 from 90 to get 54.

$$-3x+54-x^{2}=0$$

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

$$-x^{2}-3x+54=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+54. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-54=-54$$

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 -54.

$$1,-54 2,-27 3,-18 6,-9$$

Calculate the sum for each pair.

$$1-54=-53 2-27=-25 3-18=-15 6-9=-3$$

The solution is the pair that gives sum -3.

$$a=6 b=-9$$

Rewrite -x^{2}-3x+54 as \left(-x^{2}+6x\right)+\left(-9x+54\right).

$$\left(-x^{2}+6x\right)+\left(-9x+54\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(-x+6\right)+9\left(-x+6\right)$$

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

$$\left(-x+6\right)\left(x+9\right)$$

To find equation solutions, solve -x+6=0 and x+9=0.

$$x=6 x=-9$$

Substitute 6 for x in the equation 3\sqrt{x+10}=x+6.

$$3\sqrt{6+10}=6+6$$

Simplify. The value x=6 satisfies the equation.

$$12=12$$

Substitute -9 for x in the equation 3\sqrt{x+10}=x+6.

$$3\sqrt{-9+10}=-9+6$$

Simplify. The value x=-9 does not satisfy the equation because the left and the right hand side have opposite signs.

$$3=-3$$

Equation 3\sqrt{x+10}=x+6 has a unique solution.

$$x=6$$