Solve x^2-1/3-x+3/5=x-2

$$\frac { x ^ { 2 } - 1 } { 3 } - \frac { x + 3 } { 5 } = x - 2$$

Solve for x

$$x=2$$

$$x = \frac{8}{5} = 1\frac{3}{5} = 1.6$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 15, the least common multiple of 3,5.

$$5\left(x^{2}-1\right)-3\left(x+3\right)=15x-30$$

Use the distributive property to multiply 5 by x^{2}-1.

$$5x^{2}-5-3\left(x+3\right)=15x-30$$

Use the distributive property to multiply -3 by x+3.

$$5x^{2}-5-3x-9=15x-30$$

Subtract 9 from -5 to get -14.

$$5x^{2}-14-3x=15x-30$$

Subtract 15x from both sides.

$$5x^{2}-14-3x-15x=-30$$

Combine -3x and -15x to get -18x.

$$5x^{2}-14-18x=-30$$

Add 30 to both sides.

$$5x^{2}-14-18x+30=0$$

Add -14 and 30 to get 16.

$$5x^{2}+16-18x=0$$

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

$$5x^{2}-18x+16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx+16. To find a and b, set up a system to be solved.

$$a+b=-18 ab=5\times 16=80$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 80.

$$-1,-80 -2,-40 -4,-20 -5,-16 -8,-10$$

Calculate the sum for each pair.

$$-1-80=-81 -2-40=-42 -4-20=-24 -5-16=-21 -8-10=-18$$

The solution is the pair that gives sum -18.

$$a=-10 b=-8$$

Rewrite 5x^{2}-18x+16 as \left(5x^{2}-10x\right)+\left(-8x+16\right).

$$\left(5x^{2}-10x\right)+\left(-8x+16\right)$$

Factor out 5x in the first and -8 in the second group.

$$5x\left(x-2\right)-8\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(5x-8\right)$$

To find equation solutions, solve x-2=0 and 5x-8=0.

$$x=2 x=\frac{8}{5}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 15, the least common multiple of 3,5.

$$5\left(x^{2}-1\right)-3\left(x+3\right)=15x-30$$

Use the distributive property to multiply 5 by x^{2}-1.

$$5x^{2}-5-3\left(x+3\right)=15x-30$$

Use the distributive property to multiply -3 by x+3.

$$5x^{2}-5-3x-9=15x-30$$

Subtract 9 from -5 to get -14.

$$5x^{2}-14-3x=15x-30$$

Subtract 15x from both sides.

$$5x^{2}-14-3x-15x=-30$$

Combine -3x and -15x to get -18x.

$$5x^{2}-14-18x=-30$$

Add 30 to both sides.

$$5x^{2}-14-18x+30=0$$

Add -14 and 30 to get 16.

$$5x^{2}+16-18x=0$$

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.

$$5x^{2}-18x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -18 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 5\times 16}}{2\times 5}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 5\times 16}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-18\right)±\sqrt{324-20\times 16}}{2\times 5}$$

Multiply -20 times 16.

$$x=\frac{-\left(-18\right)±\sqrt{324-320}}{2\times 5}$$

Add 324 to -320.

$$x=\frac{-\left(-18\right)±\sqrt{4}}{2\times 5}$$

Take the square root of 4.

$$x=\frac{-\left(-18\right)±2}{2\times 5}$$

The opposite of -18 is 18.

$$x=\frac{18±2}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{18±2}{10}$$

Now solve the equation x=\frac{18±2}{10} when ± is plus. Add 18 to 2.

$$x=\frac{20}{10}$$

Divide 20 by 10.

$$x=2$$

Now solve the equation x=\frac{18±2}{10} when ± is minus. Subtract 2 from 18.

$$x=\frac{16}{10}$$

Reduce the fraction \frac{16}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{8}{5}$$

The equation is now solved.

$$x=2 x=\frac{8}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 15, the least common multiple of 3,5.

$$5\left(x^{2}-1\right)-3\left(x+3\right)=15x-30$$

Use the distributive property to multiply 5 by x^{2}-1.

$$5x^{2}-5-3\left(x+3\right)=15x-30$$

Use the distributive property to multiply -3 by x+3.

$$5x^{2}-5-3x-9=15x-30$$

Subtract 9 from -5 to get -14.

$$5x^{2}-14-3x=15x-30$$

Subtract 15x from both sides.

$$5x^{2}-14-3x-15x=-30$$

Combine -3x and -15x to get -18x.

$$5x^{2}-14-18x=-30$$

Add 14 to both sides.

$$5x^{2}-18x=-30+14$$

Add -30 and 14 to get -16.

$$5x^{2}-18x=-16$$

Divide both sides by 5.

$$\frac{5x^{2}-18x}{5}=-\frac{16}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{18}{5}x=-\frac{16}{5}$$

Divide -\frac{18}{5}, the coefficient of the x term, by 2 to get -\frac{9}{5}. Then add the square of -\frac{9}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{18}{5}x+\left(-\frac{9}{5}\right)^{2}=-\frac{16}{5}+\left(-\frac{9}{5}\right)^{2}$$

Square -\frac{9}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=-\frac{16}{5}+\frac{81}{25}$$

Add -\frac{16}{5} to \frac{81}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=\frac{1}{25}$$

Factor x^{2}-\frac{18}{5}x+\frac{81}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{5}\right)^{2}=\frac{1}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{5}\right)^{2}}=\sqrt{\frac{1}{25}}$$

Simplify.

$$x-\frac{9}{5}=\frac{1}{5} x-\frac{9}{5}=-\frac{1}{5}$$

Add \frac{9}{5} to both sides of the equation.

$$x=2 x=\frac{8}{5}$$