Solve 3x=frac{sqrt{x^2-36}*10}{2}

$$3 x = \frac { \sqrt { x ^ { 2 } - 36 } \cdot 10 } { 2 }$$

Solve for x

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

Solution Steps

Multiply both sides of the equation by 2.

$$6x=10\sqrt{x^{2}-36}$$

Subtract 10\sqrt{x^{2}-36} from both sides.

$$6x-10\sqrt{x^{2}-36}=0$$

Subtract 6x from both sides of the equation.

$$-10\sqrt{x^{2}-36}=-6x$$

Square both sides of the equation.

$$\left(-10\sqrt{x^{2}-36}\right)^{2}=\left(-6x\right)^{2}$$

Expand \left(-10\sqrt{x^{2}-36}\right)^{2}.

$$\left(-10\right)^{2}\left(\sqrt{x^{2}-36}\right)^{2}=\left(-6x\right)^{2}$$

Calculate -10 to the power of 2 and get 100.

$$100\left(\sqrt{x^{2}-36}\right)^{2}=\left(-6x\right)^{2}$$

Calculate \sqrt{x^{2}-36} to the power of 2 and get x^{2}-36.

$$100\left(x^{2}-36\right)=\left(-6x\right)^{2}$$

Use the distributive property to multiply 100 by x^{2}-36.

$$100x^{2}-3600=\left(-6x\right)^{2}$$

Expand \left(-6x\right)^{2}.

$$100x^{2}-3600=\left(-6\right)^{2}x^{2}$$

Calculate -6 to the power of 2 and get 36.

$$100x^{2}-3600=36x^{2}$$

Subtract 36x^{2} from both sides.

$$100x^{2}-3600-36x^{2}=0$$

Combine 100x^{2} and -36x^{2} to get 64x^{2}.

$$64x^{2}-3600=0$$

Divide both sides by 16.

$$4x^{2}-225=0$$

Consider 4x^{2}-225. Rewrite 4x^{2}-225 as \left(2x\right)^{2}-15^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2x-15\right)\left(2x+15\right)=0$$

To find equation solutions, solve 2x-15=0 and 2x+15=0.

$$x=\frac{15}{2} x=-\frac{15}{2}$$

Substitute \frac{15}{2} for x in the equation 3x=\frac{10\sqrt{x^{2}-36}}{2}.

$$3\times \frac{15}{2}=\frac{10\sqrt{\left(\frac{15}{2}\right)^{2}-36}}{2}$$

Simplify. The value x=\frac{15}{2} satisfies the equation.

$$\frac{45}{2}=\frac{45}{2}$$

Substitute -\frac{15}{2} for x in the equation 3x=\frac{10\sqrt{x^{2}-36}}{2}.

$$3\left(-\frac{15}{2}\right)=\frac{10\sqrt{\left(-\frac{15}{2}\right)^{2}-36}}{2}$$

Simplify. The value x=-\frac{15}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{45}{2}=\frac{45}{2}$$

Equation -10\sqrt{x^{2}-36}=-6x has a unique solution.

$$x=\frac{15}{2}$$