Solve x^2+16x/13+40x/9=0

$$x ^ { 2 } + \frac { 16 x } { 13 } + \frac { 40 x } { 9 } = 0$$

Solve for x

$$x = -\frac{664}{117} = -5\frac{79}{117} \approx -5.675213675$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 117, the least common multiple of 13,9.

$$117x^{2}+9\times 16x+13\times 40x=0$$

Multiply 9 and 16 to get 144.

$$117x^{2}+144x+13\times 40x=0$$

Multiply 13 and 40 to get 520.

$$117x^{2}+144x+520x=0$$

Combine 144x and 520x to get 664x.

$$117x^{2}+664x=0$$

Factor out x.

$$x\left(117x+664\right)=0$$

To find equation solutions, solve x=0 and 117x+664=0.

$$x=0 x=-\frac{664}{117}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 117, the least common multiple of 13,9.

$$117x^{2}+9\times 16x+13\times 40x=0$$

Multiply 9 and 16 to get 144.

$$117x^{2}+144x+13\times 40x=0$$

Multiply 13 and 40 to get 520.

$$117x^{2}+144x+520x=0$$

Combine 144x and 520x to get 664x.

$$117x^{2}+664x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 117 for a, 664 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-664±\sqrt{664^{2}}}{2\times 117}$$

Take the square root of 664^{2}.

$$x=\frac{-664±664}{2\times 117}$$

Multiply 2 times 117.

$$x=\frac{-664±664}{234}$$

Now solve the equation x=\frac{-664±664}{234} when ± is plus. Add -664 to 664.

$$x=\frac{0}{234}$$

Divide 0 by 234.

$$x=0$$

Now solve the equation x=\frac{-664±664}{234} when ± is minus. Subtract 664 from -664.

$$x=-\frac{1328}{234}$$

Reduce the fraction \frac{-1328}{234} to lowest terms by extracting and canceling out 2.

$$x=-\frac{664}{117}$$

The equation is now solved.

$$x=0 x=-\frac{664}{117}$$

Steps for Completing the Square

Multiply both sides of the equation by 117, the least common multiple of 13,9.

$$117x^{2}+9\times 16x+13\times 40x=0$$

Multiply 9 and 16 to get 144.

$$117x^{2}+144x+13\times 40x=0$$

Multiply 13 and 40 to get 520.

$$117x^{2}+144x+520x=0$$

Combine 144x and 520x to get 664x.

$$117x^{2}+664x=0$$

Divide both sides by 117.

$$\frac{117x^{2}+664x}{117}=\frac{0}{117}$$

Dividing by 117 undoes the multiplication by 117.

$$x^{2}+\frac{664}{117}x=\frac{0}{117}$$

Divide 0 by 117.

$$x^{2}+\frac{664}{117}x=0$$

Divide \frac{664}{117}, the coefficient of the x term, by 2 to get \frac{332}{117}. Then add the square of \frac{332}{117} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{664}{117}x+\left(\frac{332}{117}\right)^{2}=\left(\frac{332}{117}\right)^{2}$$

Square \frac{332}{117} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{664}{117}x+\frac{110224}{13689}=\frac{110224}{13689}$$

Factor x^{2}+\frac{664}{117}x+\frac{110224}{13689}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{332}{117}\right)^{2}=\frac{110224}{13689}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{332}{117}\right)^{2}}=\sqrt{\frac{110224}{13689}}$$

Simplify.

$$x+\frac{332}{117}=\frac{332}{117} x+\frac{332}{117}=-\frac{332}{117}$$

Subtract \frac{332}{117} from both sides of the equation.

$$x=0 x=-\frac{664}{117}$$