Solve 2x-9/x+4=3x-6/x+4

$$2 x - \frac { 9 } { x + 4 } = \frac { 3 x - 6 } { x + 4 }$$

Solve for x

$$x=-3$$

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

Steps Using the Quadratic Formula

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$2x\left(x+4\right)-9=3x-6$$

Use the distributive property to multiply 2x by x+4.

$$2x^{2}+8x-9=3x-6$$

Subtract 3x from both sides.

$$2x^{2}+8x-9-3x=-6$$

Combine 8x and -3x to get 5x.

$$2x^{2}+5x-9=-6$$

Add 6 to both sides.

$$2x^{2}+5x-9+6=0$$

Add -9 and 6 to get -3.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 5 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-3\right)}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-3\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\left(-3\right)}}{2\times 2}$$

Multiply -8 times -3.

$$x=\frac{-5±\sqrt{25+24}}{2\times 2}$$

Add 25 to 24.

$$x=\frac{-5±\sqrt{49}}{2\times 2}$$

Take the square root of 49.

$$x=\frac{-5±7}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±7}{4}$$

Now solve the equation x=\frac{-5±7}{4} when ± is plus. Add -5 to 7.

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

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{-5±7}{4} when ± is minus. Subtract 7 from -5.

$$x=-\frac{12}{4}$$

Divide -12 by 4.

$$x=-3$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$2x\left(x+4\right)-9=3x-6$$

Use the distributive property to multiply 2x by x+4.

$$2x^{2}+8x-9=3x-6$$

Subtract 3x from both sides.

$$2x^{2}+8x-9-3x=-6$$

Combine 8x and -3x to get 5x.

$$2x^{2}+5x-9=-6$$

Add 9 to both sides.

$$2x^{2}+5x=-6+9$$

Add -6 and 9 to get 3.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=\frac{3}{2}+\left(\frac{5}{4}\right)^{2}$$

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{3}{2}+\frac{25}{16}$$

Add \frac{3}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{49}{16}$$

Factor x^{2}+\frac{5}{2}x+\frac{25}{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}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x+\frac{5}{4}=\frac{7}{4} x+\frac{5}{4}=-\frac{7}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

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