Solve x+1/2x^2+3x-5+3/x^2+x-2=0

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

Solve for x

$$x=\frac{\sqrt{13}-9}{2}\approx -2.697224362$$

$$x=\frac{-\sqrt{13}-9}{2}\approx -6.302775638$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{5}{2},-2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right)\left(2x+5\right), the least common multiple of 2x^{2}+3x-5,x^{2}+x-2.

$$\left(x+2\right)\left(x+1\right)+\left(2x+5\right)\times 3=0$$

Use the distributive property to multiply x+2 by x+1 and combine like terms.

$$x^{2}+3x+2+\left(2x+5\right)\times 3=0$$

Use the distributive property to multiply 2x+5 by 3.

$$x^{2}+3x+2+6x+15=0$$

Combine 3x and 6x to get 9x.

$$x^{2}+9x+2+15=0$$

Add 2 and 15 to get 17.

$$x^{2}+9x+17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and 17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 17}}{2}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\times 17}}{2}$$

Multiply -4 times 17.

$$x=\frac{-9±\sqrt{81-68}}{2}$$

Add 81 to -68.

$$x=\frac{-9±\sqrt{13}}{2}$$

Now solve the equation x=\frac{-9±\sqrt{13}}{2} when ± is plus. Add -9 to \sqrt{13}.

$$x=\frac{\sqrt{13}-9}{2}$$

Now solve the equation x=\frac{-9±\sqrt{13}}{2} when ± is minus. Subtract \sqrt{13} from -9.

$$x=\frac{-\sqrt{13}-9}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{13}-9}{2} x=\frac{-\sqrt{13}-9}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{5}{2},-2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right)\left(2x+5\right), the least common multiple of 2x^{2}+3x-5,x^{2}+x-2.

$$\left(x+2\right)\left(x+1\right)+\left(2x+5\right)\times 3=0$$

Use the distributive property to multiply x+2 by x+1 and combine like terms.

$$x^{2}+3x+2+\left(2x+5\right)\times 3=0$$

Use the distributive property to multiply 2x+5 by 3.

$$x^{2}+3x+2+6x+15=0$$

Combine 3x and 6x to get 9x.

$$x^{2}+9x+2+15=0$$

Add 2 and 15 to get 17.

$$x^{2}+9x+17=0$$

Subtract 17 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+9x=-17$$

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-17+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=-17+\frac{81}{4}$$

Add -17 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{13}{4}$$

Factor x^{2}+9x+\frac{81}{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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{13}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{13}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{\sqrt{13}}{2} x+\frac{9}{2}=-\frac{\sqrt{13}}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=\frac{\sqrt{13}-9}{2} x=\frac{-\sqrt{13}-9}{2}$$