Solve 9sqrt{x-2}=2x

$$9 \sqrt { x - 2 } = 2 x$$

Solve for x

$$x=18$$

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

Solution Steps

Square both sides of the equation.

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

Expand \left(9\sqrt{x-2}\right)^{2}.

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

Calculate 9 to the power of 2 and get 81.

$$81\left(\sqrt{x-2}\right)^{2}=\left(2x\right)^{2}$$

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

$$81\left(x-2\right)=\left(2x\right)^{2}$$

Use the distributive property to multiply 81 by x-2.

$$81x-162=\left(2x\right)^{2}$$

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

$$81x-162=2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$81x-162=4x^{2}$$

Subtract 4x^{2} from both sides.

$$81x-162-4x^{2}=0$$

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

$$-4x^{2}+81x-162=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-162. To find a and b, set up a system to be solved.

$$a+b=81 ab=-4\left(-162\right)=648$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 648.

$$1,648 2,324 3,216 4,162 6,108 8,81 9,72 12,54 18,36 24,27$$

Calculate the sum for each pair.

$$1+648=649 2+324=326 3+216=219 4+162=166 6+108=114 8+81=89 9+72=81 12+54=66 18+36=54 24+27=51$$

The solution is the pair that gives sum 81.

$$a=72 b=9$$

Rewrite -4x^{2}+81x-162 as \left(-4x^{2}+72x\right)+\left(9x-162\right).

$$\left(-4x^{2}+72x\right)+\left(9x-162\right)$$

Factor out 4x in the first and -9 in the second group.

$$4x\left(-x+18\right)-9\left(-x+18\right)$$

Factor out common term -x+18 by using distributive property.

$$\left(-x+18\right)\left(4x-9\right)$$

To find equation solutions, solve -x+18=0 and 4x-9=0.

$$x=18 x=\frac{9}{4}$$

Substitute 18 for x in the equation 9\sqrt{x-2}=2x.

$$9\sqrt{18-2}=2\times 18$$

Simplify. The value x=18 satisfies the equation.

$$36=36$$

Substitute \frac{9}{4} for x in the equation 9\sqrt{x-2}=2x.

$$9\sqrt{\frac{9}{4}-2}=2\times \frac{9}{4}$$

Simplify. The value x=\frac{9}{4} satisfies the equation.

$$\frac{9}{2}=\frac{9}{2}$$

List all solutions of 9\sqrt{x-2}=2x.

$$x=18 x=\frac{9}{4}$$