Solve -4/3x^2+8/3x=x/4

$$- \frac { 4 } { 3 } x ^ { 2 } + \frac { 8 } { 3 } x = \frac { x } { 4 }$$

Solve for x

$$x = \frac{29}{16} = 1\frac{13}{16} = 1.8125$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$-16x^{2}+32x=3x$$

Subtract 3x from both sides.

$$-16x^{2}+32x-3x=0$$

Combine 32x and -3x to get 29x.

$$-16x^{2}+29x=0$$

Factor out x.

$$x\left(-16x+29\right)=0$$

To find equation solutions, solve x=0 and -16x+29=0.

$$x=0 x=\frac{29}{16}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$-16x^{2}+32x=3x$$

Subtract 3x from both sides.

$$-16x^{2}+32x-3x=0$$

Combine 32x and -3x to get 29x.

$$-16x^{2}+29x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 29 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-29±\sqrt{29^{2}}}{2\left(-16\right)}$$

Take the square root of 29^{2}.

$$x=\frac{-29±29}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-29±29}{-32}$$

Now solve the equation x=\frac{-29±29}{-32} when ± is plus. Add -29 to 29.

$$x=\frac{0}{-32}$$

Divide 0 by -32.

$$x=0$$

Now solve the equation x=\frac{-29±29}{-32} when ± is minus. Subtract 29 from -29.

$$x=-\frac{58}{-32}$$

Reduce the fraction \frac{-58}{-32} to lowest terms by extracting and canceling out 2.

$$x=\frac{29}{16}$$

The equation is now solved.

$$x=0 x=\frac{29}{16}$$

Steps for Completing the Square

Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$-16x^{2}+32x=3x$$

Subtract 3x from both sides.

$$-16x^{2}+32x-3x=0$$

Combine 32x and -3x to get 29x.

$$-16x^{2}+29x=0$$

Divide both sides by -16.

$$\frac{-16x^{2}+29x}{-16}=\frac{0}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{29}{-16}x=\frac{0}{-16}$$

Divide 29 by -16.

$$x^{2}-\frac{29}{16}x=\frac{0}{-16}$$

Divide 0 by -16.

$$x^{2}-\frac{29}{16}x=0$$

Divide -\frac{29}{16}, the coefficient of the x term, by 2 to get -\frac{29}{32}. Then add the square of -\frac{29}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{29}{16}x+\left(-\frac{29}{32}\right)^{2}=\left(-\frac{29}{32}\right)^{2}$$

Square -\frac{29}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{29}{16}x+\frac{841}{1024}=\frac{841}{1024}$$

Factor x^{2}-\frac{29}{16}x+\frac{841}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{32}\right)^{2}=\frac{841}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{32}\right)^{2}}=\sqrt{\frac{841}{1024}}$$

Simplify.

$$x-\frac{29}{32}=\frac{29}{32} x-\frac{29}{32}=-\frac{29}{32}$$

Add \frac{29}{32} to both sides of the equation.

$$x=\frac{29}{16} x=0$$