Solve -2x+7x-9=14x^2-9x-14

$$- 2 x + 7 x - 9 = 14 x ^ { 2 } - 9 x - 14$$

Solve for x

$$x=\frac{\sqrt{119}}{14}+\frac{1}{2}\approx 1.279193722$$

$$x=-\frac{\sqrt{119}}{14}+\frac{1}{2}\approx -0.279193722$$

Steps Using the Quadratic Formula

Combine -2x and 7x to get 5x.

$$5x-9=14x^{2}-9x-14$$

Subtract 14x^{2} from both sides.

$$5x-9-14x^{2}=-9x-14$$

Add 9x to both sides.

$$5x-9-14x^{2}+9x=-14$$

Combine 5x and 9x to get 14x.

$$14x-9-14x^{2}=-14$$

Add 14 to both sides.

$$14x-9-14x^{2}+14=0$$

Add -9 and 14 to get 5.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -14 for a, 14 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-14\right)\times 5}}{2\left(-14\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-14\right)\times 5}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-14±\sqrt{196+56\times 5}}{2\left(-14\right)}$$

Multiply 56 times 5.

$$x=\frac{-14±\sqrt{196+280}}{2\left(-14\right)}$$

Add 196 to 280.

$$x=\frac{-14±\sqrt{476}}{2\left(-14\right)}$$

Take the square root of 476.

$$x=\frac{-14±2\sqrt{119}}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{-14±2\sqrt{119}}{-28}$$

Now solve the equation x=\frac{-14±2\sqrt{119}}{-28} when ± is plus. Add -14 to 2\sqrt{119}.

$$x=\frac{2\sqrt{119}-14}{-28}$$

Divide -14+2\sqrt{119} by -28.

$$x=-\frac{\sqrt{119}}{14}+\frac{1}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{119}}{-28} when ± is minus. Subtract 2\sqrt{119} from -14.

$$x=\frac{-2\sqrt{119}-14}{-28}$$

Divide -14-2\sqrt{119} by -28.

$$x=\frac{\sqrt{119}}{14}+\frac{1}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{119}}{14}+\frac{1}{2} x=\frac{\sqrt{119}}{14}+\frac{1}{2}$$

Steps for Completing the Square

Combine -2x and 7x to get 5x.

$$5x-9=14x^{2}-9x-14$$

Subtract 14x^{2} from both sides.

$$5x-9-14x^{2}=-9x-14$$

Add 9x to both sides.

$$5x-9-14x^{2}+9x=-14$$

Combine 5x and 9x to get 14x.

$$14x-9-14x^{2}=-14$$

Add 9 to both sides.

$$14x-14x^{2}=-14+9$$

Add -14 and 9 to get -5.

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

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.

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

Divide both sides by -14.

$$\frac{-14x^{2}+14x}{-14}=-\frac{5}{-14}$$

Dividing by -14 undoes the multiplication by -14.

$$x^{2}+\frac{14}{-14}x=-\frac{5}{-14}$$

Divide 14 by -14.

$$x^{2}-x=-\frac{5}{-14}$$

Divide -5 by -14.

$$x^{2}-x=\frac{5}{14}$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{5}{14}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=\frac{5}{14}+\frac{1}{4}$$

Add \frac{5}{14} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{17}{28}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{17}{28}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{17}{28}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{119}}{14} x-\frac{1}{2}=-\frac{\sqrt{119}}{14}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{119}}{14}+\frac{1}{2} x=-\frac{\sqrt{119}}{14}+\frac{1}{2}$$