Solve x-2/2+1/3x^2-2/5x+1/5=frac{x^2-1}{15}

$$\frac { x - 2 } { 2 } + \frac { 1 } { 3 } x ^ { 2 } - \frac { 2 } { 5 } x + \frac { 1 } { 5 } = \frac { x ^ { 2 } - 1 } { 15 }$$

Solve for x

$$x = \frac{\sqrt{713} - 3}{16} \approx 1.48137874$$

$$x=\frac{-\sqrt{713}-3}{16}\approx -1.85637874$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 30, the least common multiple of 2,3,5,15.

$$15\left(x-2\right)+10x^{2}-12x+6=2\left(x^{2}-1\right)$$

Use the distributive property to multiply 15 by x-2.

$$15x-30+10x^{2}-12x+6=2\left(x^{2}-1\right)$$

Combine 15x and -12x to get 3x.

$$3x-30+10x^{2}+6=2\left(x^{2}-1\right)$$

Add -30 and 6 to get -24.

$$3x-24+10x^{2}=2\left(x^{2}-1\right)$$

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

$$3x-24+10x^{2}=2x^{2}-2$$

Subtract 2x^{2} from both sides.

$$3x-24+10x^{2}-2x^{2}=-2$$

Combine 10x^{2} and -2x^{2} to get 8x^{2}.

$$3x-24+8x^{2}=-2$$

Add 2 to both sides.

$$3x-24+8x^{2}+2=0$$

Add -24 and 2 to get -22.

$$3x-22+8x^{2}=0$$

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.

$$8x^{2}+3x-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 3 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 8\left(-22\right)}}{2\times 8}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 8\left(-22\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-3±\sqrt{9-32\left(-22\right)}}{2\times 8}$$

Multiply -32 times -22.

$$x=\frac{-3±\sqrt{9+704}}{2\times 8}$$

Add 9 to 704.

$$x=\frac{-3±\sqrt{713}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-3±\sqrt{713}}{16}$$

Now solve the equation x=\frac{-3±\sqrt{713}}{16} when ± is plus. Add -3 to \sqrt{713}.

$$x=\frac{\sqrt{713}-3}{16}$$

Now solve the equation x=\frac{-3±\sqrt{713}}{16} when ± is minus. Subtract \sqrt{713} from -3.

$$x=\frac{-\sqrt{713}-3}{16}$$

The equation is now solved.

$$x=\frac{\sqrt{713}-3}{16} x=\frac{-\sqrt{713}-3}{16}$$

Steps for Completing the Square

Multiply both sides of the equation by 30, the least common multiple of 2,3,5,15.

$$15\left(x-2\right)+10x^{2}-12x+6=2\left(x^{2}-1\right)$$

Use the distributive property to multiply 15 by x-2.

$$15x-30+10x^{2}-12x+6=2\left(x^{2}-1\right)$$

Combine 15x and -12x to get 3x.

$$3x-30+10x^{2}+6=2\left(x^{2}-1\right)$$

Add -30 and 6 to get -24.

$$3x-24+10x^{2}=2\left(x^{2}-1\right)$$

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

$$3x-24+10x^{2}=2x^{2}-2$$

Subtract 2x^{2} from both sides.

$$3x-24+10x^{2}-2x^{2}=-2$$

Combine 10x^{2} and -2x^{2} to get 8x^{2}.

$$3x-24+8x^{2}=-2$$

Add 24 to both sides.

$$3x+8x^{2}=-2+24$$

Add -2 and 24 to get 22.

$$3x+8x^{2}=22$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$8x^{2}+3x=22$$

Divide both sides by 8.

$$\frac{8x^{2}+3x}{8}=\frac{22}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{3}{8}x=\frac{22}{8}$$

Reduce the fraction \frac{22}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{3}{8}x=\frac{11}{4}$$

Divide \frac{3}{8}, the coefficient of the x term, by 2 to get \frac{3}{16}. Then add the square of \frac{3}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{8}x+\left(\frac{3}{16}\right)^{2}=\frac{11}{4}+\left(\frac{3}{16}\right)^{2}$$

Square \frac{3}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{8}x+\frac{9}{256}=\frac{11}{4}+\frac{9}{256}$$

Add \frac{11}{4} to \frac{9}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{8}x+\frac{9}{256}=\frac{713}{256}$$

Factor x^{2}+\frac{3}{8}x+\frac{9}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{16}\right)^{2}=\frac{713}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{16}\right)^{2}}=\sqrt{\frac{713}{256}}$$

Simplify.

$$x+\frac{3}{16}=\frac{\sqrt{713}}{16} x+\frac{3}{16}=-\frac{\sqrt{713}}{16}$$

Subtract \frac{3}{16} from both sides of the equation.

$$x=\frac{\sqrt{713}-3}{16} x=\frac{-\sqrt{713}-3}{16}$$