Solve 3x=2sqrt{(x+16)(2x-15)}

$$3x = 2 \sqrt{ (x+16)(2x-15) }$$

Solve for x

$$x=48$$

$$x=20$$

Solution Steps

Square both sides of the equation.

$$\left(3x\right)^{2}=\left(2\sqrt{\left(x+16\right)\left(2x-15\right)}\right)^{2}$$

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

$$3^{2}x^{2}=\left(2\sqrt{\left(x+16\right)\left(2x-15\right)}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9x^{2}=\left(2\sqrt{\left(x+16\right)\left(2x-15\right)}\right)^{2}$$

Use the distributive property to multiply x+16 by 2x-15 and combine like terms.

$$9x^{2}=\left(2\sqrt{2x^{2}+17x-240}\right)^{2}$$

Expand \left(2\sqrt{2x^{2}+17x-240}\right)^{2}.

$$9x^{2}=2^{2}\left(\sqrt{2x^{2}+17x-240}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$9x^{2}=4\left(\sqrt{2x^{2}+17x-240}\right)^{2}$$

Calculate \sqrt{2x^{2}+17x-240} to the power of 2 and get 2x^{2}+17x-240.

$$9x^{2}=4\left(2x^{2}+17x-240\right)$$

Use the distributive property to multiply 4 by 2x^{2}+17x-240.

$$9x^{2}=8x^{2}+68x-960$$

Subtract 8x^{2} from both sides.

$$9x^{2}-8x^{2}=68x-960$$

Combine 9x^{2} and -8x^{2} to get x^{2}.

$$x^{2}=68x-960$$

Subtract 68x from both sides.

$$x^{2}-68x=-960$$

Add 960 to both sides.

$$x^{2}-68x+960=0$$

To solve the equation, factor x^{2}-68x+960 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.

$$a+b=-68 ab=960$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 960.

$$-1,-960 -2,-480 -3,-320 -4,-240 -5,-192 -6,-160 -8,-120 -10,-96 -12,-80 -15,-64 -16,-60 -20,-48 -24,-40 -30,-32$$

Calculate the sum for each pair.

$$-1-960=-961 -2-480=-482 -3-320=-323 -4-240=-244 -5-192=-197 -6-160=-166 -8-120=-128 -10-96=-106 -12-80=-92 -15-64=-79 -16-60=-76 -20-48=-68 -24-40=-64 -30-32=-62$$

The solution is the pair that gives sum -68.

$$a=-48 b=-20$$

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

$$\left(x-48\right)\left(x-20\right)$$

To find equation solutions, solve x-48=0 and x-20=0.

$$x=48 x=20$$

Substitute 48 for x in the equation 3x=2\sqrt{\left(x+16\right)\left(2x-15\right)}.

$$3\times 48=2\sqrt{\left(48+16\right)\left(2\times 48-15\right)}$$

Simplify. The value x=48 satisfies the equation.

$$144=144$$

Substitute 20 for x in the equation 3x=2\sqrt{\left(x+16\right)\left(2x-15\right)}.

$$3\times 20=2\sqrt{\left(20+16\right)\left(2\times 20-15\right)}$$

Simplify. The value x=20 satisfies the equation.

$$60=60$$

List all solutions of 3x=2\sqrt{\left(2x-15\right)\left(x+16\right)}.

$$x=48 x=20$$