Solve 3x^2-1=1/5(1-x)

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

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=\frac{3}{5}=0.6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{5} by 1-x.

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

Subtract \frac{1}{5} from both sides.

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

Subtract \frac{1}{5} from -1 to get -\frac{6}{5}.

$$3x^{2}-\frac{6}{5}=-\frac{1}{5}x$$

Add \frac{1}{5}x to both sides.

$$3x^{2}-\frac{6}{5}+\frac{1}{5}x=0$$

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.

$$3x^{2}+\frac{1}{5}x-\frac{6}{5}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, \frac{1}{5} for b, and -\frac{6}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{5}±\sqrt{\left(\frac{1}{5}\right)^{2}-4\times 3\left(-\frac{6}{5}\right)}}{2\times 3}$$

Square \frac{1}{5} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{1}{5}±\sqrt{\frac{1}{25}-4\times 3\left(-\frac{6}{5}\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\frac{1}{5}±\sqrt{\frac{1}{25}-12\left(-\frac{6}{5}\right)}}{2\times 3}$$

Multiply -12 times -\frac{6}{5}.

$$x=\frac{-\frac{1}{5}±\sqrt{\frac{1}{25}+\frac{72}{5}}}{2\times 3}$$

Add \frac{1}{25} to \frac{72}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1}{5}±\sqrt{\frac{361}{25}}}{2\times 3}$$

Take the square root of \frac{361}{25}.

$$x=\frac{-\frac{1}{5}±\frac{19}{5}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-\frac{1}{5}±\frac{19}{5}}{6}$$

Now solve the equation x=\frac{-\frac{1}{5}±\frac{19}{5}}{6} when ± is plus. Add -\frac{1}{5} to \frac{19}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{18}{5}}{6}$$

Divide \frac{18}{5} by 6.

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

Now solve the equation x=\frac{-\frac{1}{5}±\frac{19}{5}}{6} when ± is minus. Subtract \frac{19}{5} from -\frac{1}{5} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{4}{6}$$

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{3}$$

The equation is now solved.

$$x=\frac{3}{5} x=-\frac{2}{3}$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{5} by 1-x.

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

Add \frac{1}{5}x to both sides.

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

Add 1 to both sides.

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

Add \frac{1}{5} and 1 to get \frac{6}{5}.

$$3x^{2}+\frac{1}{5}x=\frac{6}{5}$$

Divide both sides by 3.

$$\frac{3x^{2}+\frac{1}{5}x}{3}=\frac{\frac{6}{5}}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{\frac{1}{5}}{3}x=\frac{\frac{6}{5}}{3}$$

Divide \frac{1}{5} by 3.

$$x^{2}+\frac{1}{15}x=\frac{\frac{6}{5}}{3}$$

Divide \frac{6}{5} by 3.

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

Divide \frac{1}{15}, the coefficient of the x term, by 2 to get \frac{1}{30}. Then add the square of \frac{1}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{1}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{15}x+\frac{1}{900}=\frac{2}{5}+\frac{1}{900}$$

Add \frac{2}{5} to \frac{1}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{15}x+\frac{1}{900}=\frac{361}{900}$$

Factor x^{2}+\frac{1}{15}x+\frac{1}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{30}\right)^{2}=\frac{361}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{30}\right)^{2}}=\sqrt{\frac{361}{900}}$$

Simplify.

$$x+\frac{1}{30}=\frac{19}{30} x+\frac{1}{30}=-\frac{19}{30}$$

Subtract \frac{1}{30} from both sides of the equation.

$$x=\frac{3}{5} x=-\frac{2}{3}$$