Solve x=2sqrt{x+15}

$$x = 2 \sqrt { x + 15 }$$

Solve for x

$$x=10$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(2\sqrt{x+15}\right)^{2}$$

Expand \left(2\sqrt{x+15}\right)^{2}.

$$x^{2}=2^{2}\left(\sqrt{x+15}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}=4\left(\sqrt{x+15}\right)^{2}$$

Calculate \sqrt{x+15} to the power of 2 and get x+15.

$$x^{2}=4\left(x+15\right)$$

Use the distributive property to multiply 4 by x+15.

$$x^{2}=4x+60$$

Subtract 4x from both sides.

$$x^{2}-4x=60$$

Subtract 60 from both sides.

$$x^{2}-4x-60=0$$

To solve the equation, factor x^{2}-4x-60 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=-60$$

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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -4.

$$a=-10 b=6$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-10\right)\left(x+6\right)$$

To find equation solutions, solve x-10=0 and x+6=0.

$$x=10 x=-6$$

Substitute 10 for x in the equation x=2\sqrt{x+15}.

$$10=2\sqrt{10+15}$$

Simplify. The value x=10 satisfies the equation.

$$10=10$$

Substitute -6 for x in the equation x=2\sqrt{x+15}.

$$-6=2\sqrt{-6+15}$$

Simplify. The value x=-6 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-6=6$$

Equation x=2\sqrt{x+15} has a unique solution.

$$x=10$$