Solve 2x-1/x+3=-2x+1

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

Solve for x

$$x=-4$$

$$x=\frac{1}{2}=0.5$$

Steps Using the Quadratic Formula

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

$$2x-1=-2x\left(x+3\right)+x+3$$

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

$$2x-1=-2x^{2}-6x+x+3$$

Combine -6x and x to get -5x.

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

Add 2x^{2} to both sides.

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

Add 5x to both sides.

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

Combine 2x and 5x to get 7x.

$$7x-1+2x^{2}=3$$

Subtract 3 from both sides.

$$7x-1+2x^{2}-3=0$$

Subtract 3 from -1 to get -4.

$$7x-4+2x^{2}=0$$

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.

$$2x^{2}+7x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 7 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 2\left(-4\right)}}{2\times 2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 2\left(-4\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-7±\sqrt{49-8\left(-4\right)}}{2\times 2}$$

Multiply -8 times -4.

$$x=\frac{-7±\sqrt{49+32}}{2\times 2}$$

Add 49 to 32.

$$x=\frac{-7±\sqrt{81}}{2\times 2}$$

Take the square root of 81.

$$x=\frac{-7±9}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-7±9}{4}$$

Now solve the equation x=\frac{-7±9}{4} when ± is plus. Add -7 to 9.

$$x=\frac{2}{4}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{-7±9}{4} when ± is minus. Subtract 9 from -7.

$$x=-\frac{16}{4}$$

Divide -16 by 4.

$$x=-4$$

The equation is now solved.

$$x=\frac{1}{2} x=-4$$

Steps for Completing the Square

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

$$2x-1=-2x\left(x+3\right)+x+3$$

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

$$2x-1=-2x^{2}-6x+x+3$$

Combine -6x and x to get -5x.

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

Add 2x^{2} to both sides.

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

Add 5x to both sides.

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

Combine 2x and 5x to get 7x.

$$7x-1+2x^{2}=3$$

Add 1 to both sides.

$$7x+2x^{2}=3+1$$

Add 3 and 1 to get 4.

$$7x+2x^{2}=4$$

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.

$$2x^{2}+7x=4$$

Divide both sides by 2.

$$\frac{2x^{2}+7x}{2}=\frac{4}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{7}{2}x=\frac{4}{2}$$

Divide 4 by 2.

$$x^{2}+\frac{7}{2}x=2$$

Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=2+\left(\frac{7}{4}\right)^{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=2+\frac{49}{16}$$

Add 2 to \frac{49}{16}.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{81}{16}$$

Factor x^{2}+\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$x+\frac{7}{4}=\frac{9}{4} x+\frac{7}{4}=-\frac{9}{4}$$

Subtract \frac{7}{4} from both sides of the equation.

$$x=\frac{1}{2} x=-4$$