Solve 3sqrt{2x-5}=x-6

$$3 \sqrt { 2 x - 5 } = x - 6$$

Solve for x

$$x=27$$

Solution Steps

Square both sides of the equation.

$$\left(3\sqrt{2x-5}\right)^{2}=\left(x-6\right)^{2}$$

Expand \left(3\sqrt{2x-5}\right)^{2}.

$$3^{2}\left(\sqrt{2x-5}\right)^{2}=\left(x-6\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{2x-5}\right)^{2}=\left(x-6\right)^{2}$$

Calculate \sqrt{2x-5} to the power of 2 and get 2x-5.

$$9\left(2x-5\right)=\left(x-6\right)^{2}$$

Use the distributive property to multiply 9 by 2x-5.

$$18x-45=\left(x-6\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$18x-45=x^{2}-12x+36$$

Subtract x^{2} from both sides.

$$18x-45-x^{2}=-12x+36$$

Add 12x to both sides.

$$18x-45-x^{2}+12x=36$$

Combine 18x and 12x to get 30x.

$$30x-45-x^{2}=36$$

Subtract 36 from both sides.

$$30x-45-x^{2}-36=0$$

Subtract 36 from -45 to get -81.

$$30x-81-x^{2}=0$$

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

$$-x^{2}+30x-81=0$$

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

$$a+b=30 ab=-\left(-81\right)=81$$

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 81.

$$1,81 3,27 9,9$$

Calculate the sum for each pair.

$$1+81=82 3+27=30 9+9=18$$

The solution is the pair that gives sum 30.

$$a=27 b=3$$

Rewrite -x^{2}+30x-81 as \left(-x^{2}+27x\right)+\left(3x-81\right).

$$\left(-x^{2}+27x\right)+\left(3x-81\right)$$

Factor out -x in the first and 3 in the second group.

$$-x\left(x-27\right)+3\left(x-27\right)$$

Factor out common term x-27 by using distributive property.

$$\left(x-27\right)\left(-x+3\right)$$

To find equation solutions, solve x-27=0 and -x+3=0.

$$x=27 x=3$$

Substitute 27 for x in the equation 3\sqrt{2x-5}=x-6.

$$3\sqrt{2\times 27-5}=27-6$$

Simplify. The value x=27 satisfies the equation.

$$21=21$$

Substitute 3 for x in the equation 3\sqrt{2x-5}=x-6.

$$3\sqrt{2\times 3-5}=3-6$$

Simplify. The value x=3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$3=-3$$

Equation 3\sqrt{2x-5}=x-6 has a unique solution.

$$x=27$$