Solve 4x/x-4=3x/x^2-16

$$\frac { 4 x } { x - 4 } = \frac { 3 x } { x ^ { 2 } - 16 }$$

Solve for x

$$x = -\frac{13}{4} = -3\frac{1}{4} = -3.25$$

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to any of the values -4,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+4\right), the least common multiple of x-4,x^{2}-16.

$$\left(x+4\right)\times 4x=3x$$

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

$$\left(4x+16\right)x=3x$$

Use the distributive property to multiply 4x+16 by x.

$$4x^{2}+16x=3x$$

Subtract 3x from both sides.

$$4x^{2}+16x-3x=0$$

Combine 16x and -3x to get 13x.

$$4x^{2}+13x=0$$

Factor out x.

$$x\left(4x+13\right)=0$$

To find equation solutions, solve x=0 and 4x+13=0.

$$x=0 x=-\frac{13}{4}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+4\right), the least common multiple of x-4,x^{2}-16.

$$\left(x+4\right)\times 4x=3x$$

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

$$\left(4x+16\right)x=3x$$

Use the distributive property to multiply 4x+16 by x.

$$4x^{2}+16x=3x$$

Subtract 3x from both sides.

$$4x^{2}+16x-3x=0$$

Combine 16x and -3x to get 13x.

$$4x^{2}+13x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 13 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}}}{2\times 4}$$

Take the square root of 13^{2}.

$$x=\frac{-13±13}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-13±13}{8}$$

Now solve the equation x=\frac{-13±13}{8} when ± is plus. Add -13 to 13.

$$x=\frac{0}{8}$$

Divide 0 by 8.

$$x=0$$

Now solve the equation x=\frac{-13±13}{8} when ± is minus. Subtract 13 from -13.

$$x=-\frac{26}{8}$$

Reduce the fraction \frac{-26}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{13}{4}$$

The equation is now solved.

$$x=0 x=-\frac{13}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+4\right), the least common multiple of x-4,x^{2}-16.

$$\left(x+4\right)\times 4x=3x$$

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

$$\left(4x+16\right)x=3x$$

Use the distributive property to multiply 4x+16 by x.

$$4x^{2}+16x=3x$$

Subtract 3x from both sides.

$$4x^{2}+16x-3x=0$$

Combine 16x and -3x to get 13x.

$$4x^{2}+13x=0$$

Divide both sides by 4.

$$\frac{4x^{2}+13x}{4}=\frac{0}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{13}{4}x=\frac{0}{4}$$

Divide 0 by 4.

$$x^{2}+\frac{13}{4}x=0$$

Divide \frac{13}{4}, the coefficient of the x term, by 2 to get \frac{13}{8}. Then add the square of \frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{4}x+\left(\frac{13}{8}\right)^{2}=\left(\frac{13}{8}\right)^{2}$$

Square \frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=\frac{169}{64}$$

Factor x^{2}+\frac{13}{4}x+\frac{169}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{8}\right)^{2}=\frac{169}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Simplify.

$$x+\frac{13}{8}=\frac{13}{8} x+\frac{13}{8}=-\frac{13}{8}$$

Subtract \frac{13}{8} from both sides of the equation.

$$x=0 x=-\frac{13}{4}$$