Solve 3sqrt{x}=2x-9

$$3 \sqrt{ x } =2x-9$$

Solve for x

$$x=9$$

Solution Steps

Square both sides of the equation.

$$\left(3\sqrt{x}\right)^{2}=\left(2x-9\right)^{2}$$

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

$$3^{2}\left(\sqrt{x}\right)^{2}=\left(2x-9\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-9\right)^{2}.

$$9x=4x^{2}-36x+81$$

Subtract 4x^{2} from both sides.

$$9x-4x^{2}=-36x+81$$

Add 36x to both sides.

$$9x-4x^{2}+36x=81$$

Combine 9x and 36x to get 45x.

$$45x-4x^{2}=81$$

Subtract 81 from both sides.

$$45x-4x^{2}-81=0$$

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

$$-4x^{2}+45x-81=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-81. To find a and b, set up a system to be solved.

$$a+b=45 ab=-4\left(-81\right)=324$$

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 324.

$$1,324 2,162 3,108 4,81 6,54 9,36 12,27 18,18$$

Calculate the sum for each pair.

$$1+324=325 2+162=164 3+108=111 4+81=85 6+54=60 9+36=45 12+27=39 18+18=36$$

The solution is the pair that gives sum 45.

$$a=36 b=9$$

Rewrite -4x^{2}+45x-81 as \left(-4x^{2}+36x\right)+\left(9x-81\right).

$$\left(-4x^{2}+36x\right)+\left(9x-81\right)$$

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

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

Factor out common term -x+9 by using distributive property.

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

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

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

Substitute 9 for x in the equation 3\sqrt{x}=2x-9.

$$3\sqrt{9}=2\times 9-9$$

Simplify. The value x=9 satisfies the equation.

$$9=9$$

Substitute \frac{9}{4} for x in the equation 3\sqrt{x}=2x-9.

$$3\sqrt{\frac{9}{4}}=2\times \frac{9}{4}-9$$

Simplify. The value x=\frac{9}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

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

Equation 3\sqrt{x}=2x-9 has a unique solution.

$$x=9$$