Solve 8/46*2=2/3x^2-4/3x-2

$$\frac { 8 } { 46 } \times 2 = \frac { 2 } { 3 } x ^ { 2 } - \frac { 4 } { 3 } x - 2$$

Solve for x

$$x=\frac{2\sqrt{598}}{23}+1\approx 3.126438132$$

$$x=-\frac{2\sqrt{598}}{23}+1\approx -1.126438132$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{8}{46} to lowest terms by extracting and canceling out 2.

$$\frac{4}{23}\times 2=\frac{2}{3}x^{2}-\frac{4}{3}x-2$$

Multiply \frac{4}{23} and 2 to get \frac{8}{23}.

$$\frac{8}{23}=\frac{2}{3}x^{2}-\frac{4}{3}x-2$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{2}{3}x^{2}-\frac{4}{3}x-2=\frac{8}{23}$$

Subtract \frac{8}{23} from both sides.

$$\frac{2}{3}x^{2}-\frac{4}{3}x-2-\frac{8}{23}=0$$

Subtract \frac{8}{23} from -2 to get -\frac{54}{23}.

$$\frac{2}{3}x^{2}-\frac{4}{3}x-\frac{54}{23}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{2}{3} for a, -\frac{4}{3} for b, and -\frac{54}{23} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\left(-\frac{4}{3}\right)^{2}-4\times \frac{2}{3}\left(-\frac{54}{23}\right)}}{2\times \frac{2}{3}}$$

Square -\frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}-4\times \frac{2}{3}\left(-\frac{54}{23}\right)}}{2\times \frac{2}{3}}$$

Multiply -4 times \frac{2}{3}.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}-\frac{8}{3}\left(-\frac{54}{23}\right)}}{2\times \frac{2}{3}}$$

Multiply -\frac{8}{3} times -\frac{54}{23} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}+\frac{144}{23}}}{2\times \frac{2}{3}}$$

Add \frac{16}{9} to \frac{144}{23} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{1664}{207}}}{2\times \frac{2}{3}}$$

Take the square root of \frac{1664}{207}.

$$x=\frac{-\left(-\frac{4}{3}\right)±\frac{8\sqrt{598}}{69}}{2\times \frac{2}{3}}$$

The opposite of -\frac{4}{3} is \frac{4}{3}.

$$x=\frac{\frac{4}{3}±\frac{8\sqrt{598}}{69}}{2\times \frac{2}{3}}$$

Multiply 2 times \frac{2}{3}.

$$x=\frac{\frac{4}{3}±\frac{8\sqrt{598}}{69}}{\frac{4}{3}}$$

Now solve the equation x=\frac{\frac{4}{3}±\frac{8\sqrt{598}}{69}}{\frac{4}{3}} when ± is plus. Add \frac{4}{3} to \frac{8\sqrt{598}}{69}.

$$x=\frac{\frac{8\sqrt{598}}{69}+\frac{4}{3}}{\frac{4}{3}}$$

Divide \frac{4}{3}+\frac{8\sqrt{598}}{69} by \frac{4}{3} by multiplying \frac{4}{3}+\frac{8\sqrt{598}}{69} by the reciprocal of \frac{4}{3}.

$$x=\frac{2\sqrt{598}}{23}+1$$

Now solve the equation x=\frac{\frac{4}{3}±\frac{8\sqrt{598}}{69}}{\frac{4}{3}} when ± is minus. Subtract \frac{8\sqrt{598}}{69} from \frac{4}{3}.

$$x=\frac{-\frac{8\sqrt{598}}{69}+\frac{4}{3}}{\frac{4}{3}}$$

Divide \frac{4}{3}-\frac{8\sqrt{598}}{69} by \frac{4}{3} by multiplying \frac{4}{3}-\frac{8\sqrt{598}}{69} by the reciprocal of \frac{4}{3}.

$$x=-\frac{2\sqrt{598}}{23}+1$$

The equation is now solved.

$$x=\frac{2\sqrt{598}}{23}+1 x=-\frac{2\sqrt{598}}{23}+1$$

Steps for Completing the Square

Reduce the fraction \frac{8}{46} to lowest terms by extracting and canceling out 2.

$$\frac{4}{23}\times 2=\frac{2}{3}x^{2}-\frac{4}{3}x-2$$

Multiply \frac{4}{23} and 2 to get \frac{8}{23}.

$$\frac{8}{23}=\frac{2}{3}x^{2}-\frac{4}{3}x-2$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{2}{3}x^{2}-\frac{4}{3}x-2=\frac{8}{23}$$

Add 2 to both sides.

$$\frac{2}{3}x^{2}-\frac{4}{3}x=\frac{8}{23}+2$$

Add \frac{8}{23} and 2 to get \frac{54}{23}.

$$\frac{2}{3}x^{2}-\frac{4}{3}x=\frac{54}{23}$$

Divide both sides of the equation by \frac{2}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{2}{3}x^{2}-\frac{4}{3}x}{\frac{2}{3}}=\frac{\frac{54}{23}}{\frac{2}{3}}$$

Dividing by \frac{2}{3} undoes the multiplication by \frac{2}{3}.

$$x^{2}+\left(-\frac{\frac{4}{3}}{\frac{2}{3}}\right)x=\frac{\frac{54}{23}}{\frac{2}{3}}$$

Divide -\frac{4}{3} by \frac{2}{3} by multiplying -\frac{4}{3} by the reciprocal of \frac{2}{3}.

$$x^{2}-2x=\frac{\frac{54}{23}}{\frac{2}{3}}$$

Divide \frac{54}{23} by \frac{2}{3} by multiplying \frac{54}{23} by the reciprocal of \frac{2}{3}.

$$x^{2}-2x=\frac{81}{23}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{81}{23}+1$$

Add \frac{81}{23} to 1.

$$x^{2}-2x+1=\frac{104}{23}$$

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

$$\left(x-1\right)^{2}=\frac{104}{23}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{104}{23}}$$

Simplify.

$$x-1=\frac{2\sqrt{598}}{23} x-1=-\frac{2\sqrt{598}}{23}$$

Add 1 to both sides of the equation.

$$x=\frac{2\sqrt{598}}{23}+1 x=-\frac{2\sqrt{598}}{23}+1$$