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

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

Solve for x

$$x=2$$

Solution Steps

Subtract 2 from both sides of the equation.

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

Subtract 2 from -6 to get -8.

$$\sqrt{3x-2}=5x-8$$

Square both sides of the equation.

$$\left(\sqrt{3x-2}\right)^{2}=\left(5x-8\right)^{2}$$

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

$$3x-2=\left(5x-8\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-8\right)^{2}.

$$3x-2=25x^{2}-80x+64$$

Subtract 25x^{2} from both sides.

$$3x-2-25x^{2}=-80x+64$$

Add 80x to both sides.

$$3x-2-25x^{2}+80x=64$$

Combine 3x and 80x to get 83x.

$$83x-2-25x^{2}=64$$

Subtract 64 from both sides.

$$83x-2-25x^{2}-64=0$$

Subtract 64 from -2 to get -66.

$$83x-66-25x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-25x^{2}+83x-66=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -25 for a, 83 for b, and -66 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-83±\sqrt{83^{2}-4\left(-25\right)\left(-66\right)}}{2\left(-25\right)}$$

Square 83.

$$x=\frac{-83±\sqrt{6889-4\left(-25\right)\left(-66\right)}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-83±\sqrt{6889+100\left(-66\right)}}{2\left(-25\right)}$$

Multiply 100 times -66.

$$x=\frac{-83±\sqrt{6889-6600}}{2\left(-25\right)}$$

Add 6889 to -6600.

$$x=\frac{-83±\sqrt{289}}{2\left(-25\right)}$$

Take the square root of 289.

$$x=\frac{-83±17}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{-83±17}{-50}$$

Now solve the equation x=\frac{-83±17}{-50} when ± is plus. Add -83 to 17.

$$x=-\frac{66}{-50}$$

Reduce the fraction \frac{-66}{-50} to lowest terms by extracting and canceling out 2.

$$x=\frac{33}{25}$$

Now solve the equation x=\frac{-83±17}{-50} when ± is minus. Subtract 17 from -83.

$$x=-\frac{100}{-50}$$

Divide -100 by -50.

$$x=2$$

The equation is now solved.

$$x=\frac{33}{25} x=2$$

Substitute \frac{33}{25} for x in the equation 2+\sqrt{3x-2}=5x-6.

$$2+\sqrt{3\times \frac{33}{25}-2}=5\times \frac{33}{25}-6$$

Simplify. The value x=\frac{33}{25} does not satisfy the equation.

$$\frac{17}{5}=\frac{3}{5}$$

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

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

Simplify. The value x=2 satisfies the equation.

$$4=4$$

Equation \sqrt{3x-2}=5x-8 has a unique solution.

$$x=2$$