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

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

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

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

To raise \frac{6}{\sqrt{x+3}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Calculate 6 to the power of 2 and get 36.

$$\frac{36}{\left(\sqrt{x+3}\right)^{2}}=x+3$$

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

$$\frac{36}{x+3}=x+3$$

Multiply both sides of the equation by x+3.

$$36=\left(x+3\right)x+\left(x+3\right)\times 3$$

Use the distributive property to multiply x+3 by x.

$$36=x^{2}+3x+\left(x+3\right)\times 3$$

Use the distributive property to multiply x+3 by 3.

$$36=x^{2}+3x+3x+9$$

Combine 3x and 3x to get 6x.

$$36=x^{2}+6x+9$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+6x+9=36$$

Subtract 36 from both sides.

$$x^{2}+6x+9-36=0$$

Subtract 36 from 9 to get -27.

$$x^{2}+6x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-27\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-27\right)}}{2}$$

Multiply -4 times -27.

$$x=\frac{-6±\sqrt{36+108}}{2}$$

Add 36 to 108.

$$x=\frac{-6±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-6±12}{2}$$

Now solve the equation x=\frac{-6±12}{2} when ± is plus. Add -6 to 12.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

Now solve the equation x=\frac{-6±12}{2} when ± is minus. Subtract 12 from -6.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=3 x=-9$$

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

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

Simplify. The value x=3 satisfies the equation.

$$6^{\frac{1}{2}}=6^{\frac{1}{2}}$$

Substitute -9 for x in the equation \frac{6}{\sqrt{x+3}}=\sqrt{x+3}. The expression \sqrt{-9+3} is undefined because the radicand cannot be negative.

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

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

$$x=3$$