Solve 1-2x^2-x-15/3x=0

$$1 - \frac { 2 x ^ { 2 } - x - 15 } { 3 x } = 0$$

Solve for x

$$x=\frac{\sqrt{34}}{2}+1\approx 3.915475947$$

$$x=-\frac{\sqrt{34}}{2}+1\approx -1.915475947$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x.

$$3x-\left(2x^{2}-x-15\right)=0$$

To find the opposite of 2x^{2}-x-15, find the opposite of each term.

$$3x-2x^{2}+x+15=0$$

Combine 3x and x to get 4x.

$$4x-2x^{2}+15=0$$

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.

$$-2x^{2}+4x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 4 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-2\right)\times 15}}{2\left(-2\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)\times 15}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8\times 15}}{2\left(-2\right)}$$

Multiply 8 times 15.

$$x=\frac{-4±\sqrt{16+120}}{2\left(-2\right)}$$

Add 16 to 120.

$$x=\frac{-4±\sqrt{136}}{2\left(-2\right)}$$

Take the square root of 136.

$$x=\frac{-4±2\sqrt{34}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-4±2\sqrt{34}}{-4}$$

Now solve the equation x=\frac{-4±2\sqrt{34}}{-4} when ± is plus. Add -4 to 2\sqrt{34}.

$$x=\frac{2\sqrt{34}-4}{-4}$$

Divide -4+2\sqrt{34} by -4.

$$x=-\frac{\sqrt{34}}{2}+1$$

Now solve the equation x=\frac{-4±2\sqrt{34}}{-4} when ± is minus. Subtract 2\sqrt{34} from -4.

$$x=\frac{-2\sqrt{34}-4}{-4}$$

Divide -4-2\sqrt{34} by -4.

$$x=\frac{\sqrt{34}}{2}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{34}}{2}+1 x=\frac{\sqrt{34}}{2}+1$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x.

$$3x-\left(2x^{2}-x-15\right)=0$$

To find the opposite of 2x^{2}-x-15, find the opposite of each term.

$$3x-2x^{2}+x+15=0$$

Combine 3x and x to get 4x.

$$4x-2x^{2}+15=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$4x-2x^{2}=-15$$

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.

$$-2x^{2}+4x=-15$$

Divide both sides by -2.

$$\frac{-2x^{2}+4x}{-2}=-\frac{15}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{4}{-2}x=-\frac{15}{-2}$$

Divide 4 by -2.

$$x^{2}-2x=-\frac{15}{-2}$$

Divide -15 by -2.

$$x^{2}-2x=\frac{15}{2}$$

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.

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

Add \frac{15}{2} to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=\frac{\sqrt{34}}{2} x-1=-\frac{\sqrt{34}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{34}}{2}+1 x=-\frac{\sqrt{34}}{2}+1$$