Solve 1sqrt{3x-2}=x-4

$$1 \sqrt{ 3x-2 } = x-4$$

Solve for x

$$x=9$$

Solution Steps

Square both sides of the equation.

$$\left(1\sqrt{3x-2}\right)^{2}=\left(x-4\right)^{2}$$

Expand \left(1\sqrt{3x-2}\right)^{2}.

$$1^{2}\left(\sqrt{3x-2}\right)^{2}=\left(x-4\right)^{2}$$

Calculate 1 to the power of 2 and get 1.

$$1\left(\sqrt{3x-2}\right)^{2}=\left(x-4\right)^{2}$$

Calculate \sqrt{3x-2} to the power of 2 and get 3x-2.

$$1\left(3x-2\right)=\left(x-4\right)^{2}$$

Use the distributive property to multiply 1 by 3x-2.

$$3x-2=\left(x-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$3x-2=x^{2}-8x+16$$

Subtract x^{2} from both sides.

$$3x-2-x^{2}=-8x+16$$

Add 8x to both sides.

$$3x-2-x^{2}+8x=16$$

Combine 3x and 8x to get 11x.

$$11x-2-x^{2}=16$$

Subtract 16 from both sides.

$$11x-2-x^{2}-16=0$$

Subtract 16 from -2 to get -18.

$$11x-18-x^{2}=0$$

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

$$-x^{2}+11x-18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=11 ab=-\left(-18\right)=18$$

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 18.

$$1,18 2,9 3,6$$

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 11.

$$a=9 b=2$$

Rewrite -x^{2}+11x-18 as \left(-x^{2}+9x\right)+\left(2x-18\right).

$$\left(-x^{2}+9x\right)+\left(2x-18\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-9\right)+2\left(x-9\right)$$

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

$$\left(x-9\right)\left(-x+2\right)$$

To find equation solutions, solve x-9=0 and -x+2=0.

$$x=9 x=2$$

Substitute 9 for x in the equation 1\sqrt{3x-2}=x-4.

$$1\sqrt{3\times 9-2}=9-4$$

Simplify. The value x=9 satisfies the equation.

$$5=5$$

Substitute 2 for x in the equation 1\sqrt{3x-2}=x-4.

$$1\sqrt{3\times 2-2}=2-4$$

Simplify. The value x=2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

Equation \sqrt{3x-2}=x-4 has a unique solution.

$$x=9$$