Solve 3x^2-20x^2-77x+98/-2x^3+2x^2+2x-2=0

$$\frac { 3 x ^ { 2 } - 20 x ^ { 2 } - 77 x + 98 } { - 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x - 2 } = 0$$

Solve for x

$$x = \frac{7 \sqrt{257} - 77}{34} \approx 1.035839317$$

$$x=\frac{-7\sqrt{257}-77}{34}\approx -5.565251082$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(x+1\right)\left(-x+1\right).

$$3x^{2}-20x^{2}-77x+98=0$$

Combine 3x^{2} and -20x^{2} to get -17x^{2}.

$$-17x^{2}-77x+98=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -17 for a, -77 for b, and 98 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-77\right)±\sqrt{\left(-77\right)^{2}-4\left(-17\right)\times 98}}{2\left(-17\right)}$$

Square -77.

$$x=\frac{-\left(-77\right)±\sqrt{5929-4\left(-17\right)\times 98}}{2\left(-17\right)}$$

Multiply -4 times -17.

$$x=\frac{-\left(-77\right)±\sqrt{5929+68\times 98}}{2\left(-17\right)}$$

Multiply 68 times 98.

$$x=\frac{-\left(-77\right)±\sqrt{5929+6664}}{2\left(-17\right)}$$

Add 5929 to 6664.

$$x=\frac{-\left(-77\right)±\sqrt{12593}}{2\left(-17\right)}$$

Take the square root of 12593.

$$x=\frac{-\left(-77\right)±7\sqrt{257}}{2\left(-17\right)}$$

The opposite of -77 is 77.

$$x=\frac{77±7\sqrt{257}}{2\left(-17\right)}$$

Multiply 2 times -17.

$$x=\frac{77±7\sqrt{257}}{-34}$$

Now solve the equation x=\frac{77±7\sqrt{257}}{-34} when ± is plus. Add 77 to 7\sqrt{257}.

$$x=\frac{7\sqrt{257}+77}{-34}$$

Divide 77+7\sqrt{257} by -34.

$$x=\frac{-7\sqrt{257}-77}{34}$$

Now solve the equation x=\frac{77±7\sqrt{257}}{-34} when ± is minus. Subtract 7\sqrt{257} from 77.

$$x=\frac{77-7\sqrt{257}}{-34}$$

Divide 77-7\sqrt{257} by -34.

$$x=\frac{7\sqrt{257}-77}{34}$$

The equation is now solved.

$$x=\frac{-7\sqrt{257}-77}{34} x=\frac{7\sqrt{257}-77}{34}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(x+1\right)\left(-x+1\right).

$$3x^{2}-20x^{2}-77x+98=0$$

Combine 3x^{2} and -20x^{2} to get -17x^{2}.

$$-17x^{2}-77x+98=0$$

Subtract 98 from both sides. Anything subtracted from zero gives its negation.

$$-17x^{2}-77x=-98$$

Divide both sides by -17.

$$\frac{-17x^{2}-77x}{-17}=-\frac{98}{-17}$$

Dividing by -17 undoes the multiplication by -17.

$$x^{2}+\left(-\frac{77}{-17}\right)x=-\frac{98}{-17}$$

Divide -77 by -17.

$$x^{2}+\frac{77}{17}x=-\frac{98}{-17}$$

Divide -98 by -17.

$$x^{2}+\frac{77}{17}x=\frac{98}{17}$$

Divide \frac{77}{17}, the coefficient of the x term, by 2 to get \frac{77}{34}. Then add the square of \frac{77}{34} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{77}{17}x+\left(\frac{77}{34}\right)^{2}=\frac{98}{17}+\left(\frac{77}{34}\right)^{2}$$

Square \frac{77}{34} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{77}{17}x+\frac{5929}{1156}=\frac{98}{17}+\frac{5929}{1156}$$

Add \frac{98}{17} to \frac{5929}{1156} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{77}{17}x+\frac{5929}{1156}=\frac{12593}{1156}$$

Factor x^{2}+\frac{77}{17}x+\frac{5929}{1156}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{77}{34}\right)^{2}=\frac{12593}{1156}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{77}{34}\right)^{2}}=\sqrt{\frac{12593}{1156}}$$

Simplify.

$$x+\frac{77}{34}=\frac{7\sqrt{257}}{34} x+\frac{77}{34}=-\frac{7\sqrt{257}}{34}$$

Subtract \frac{77}{34} from both sides of the equation.

$$x=\frac{7\sqrt{257}-77}{34} x=\frac{-7\sqrt{257}-77}{34}$$