Solve 4x^2/2-33x/3+14/6=0

$$\frac { 4 x ^ { 2 } } { 2 } - \frac { 33 x } { 3 } + \frac { 14 } { 6 } = 0$$

Solve for x

$$x=\frac{\sqrt{921}}{12}+\frac{11}{4}\approx 5.278998484$$

$$x=-\frac{\sqrt{921}}{12}+\frac{11}{4}\approx 0.221001516$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 2,3,6.

$$3\times 4x^{2}-2\times 33x+14=0$$

Multiply 3 and 4 to get 12.

$$12x^{2}-2\times 33x+14=0$$

Multiply -2 and 33 to get -66.

$$12x^{2}-66x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -66 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\times 12\times 14}}{2\times 12}$$

Square -66.

$$x=\frac{-\left(-66\right)±\sqrt{4356-4\times 12\times 14}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-66\right)±\sqrt{4356-48\times 14}}{2\times 12}$$

Multiply -48 times 14.

$$x=\frac{-\left(-66\right)±\sqrt{4356-672}}{2\times 12}$$

Add 4356 to -672.

$$x=\frac{-\left(-66\right)±\sqrt{3684}}{2\times 12}$$

Take the square root of 3684.

$$x=\frac{-\left(-66\right)±2\sqrt{921}}{2\times 12}$$

The opposite of -66 is 66.

$$x=\frac{66±2\sqrt{921}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{66±2\sqrt{921}}{24}$$

Now solve the equation x=\frac{66±2\sqrt{921}}{24} when ± is plus. Add 66 to 2\sqrt{921}.

$$x=\frac{2\sqrt{921}+66}{24}$$

Divide 66+2\sqrt{921} by 24.

$$x=\frac{\sqrt{921}}{12}+\frac{11}{4}$$

Now solve the equation x=\frac{66±2\sqrt{921}}{24} when ± is minus. Subtract 2\sqrt{921} from 66.

$$x=\frac{66-2\sqrt{921}}{24}$$

Divide 66-2\sqrt{921} by 24.

$$x=-\frac{\sqrt{921}}{12}+\frac{11}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{921}}{12}+\frac{11}{4} x=-\frac{\sqrt{921}}{12}+\frac{11}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 2,3,6.

$$3\times 4x^{2}-2\times 33x+14=0$$

Multiply 3 and 4 to get 12.

$$12x^{2}-2\times 33x+14=0$$

Multiply -2 and 33 to get -66.

$$12x^{2}-66x+14=0$$

Subtract 14 from both sides. Anything subtracted from zero gives its negation.

$$12x^{2}-66x=-14$$

Divide both sides by 12.

$$\frac{12x^{2}-66x}{12}=-\frac{14}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{66}{12}\right)x=-\frac{14}{12}$$

Reduce the fraction \frac{-66}{12} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{11}{2}x=-\frac{14}{12}$$

Reduce the fraction \frac{-14}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{11}{2}x=-\frac{7}{6}$$

Divide -\frac{11}{2}, the coefficient of the x term, by 2 to get -\frac{11}{4}. Then add the square of -\frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-\frac{7}{6}+\left(-\frac{11}{4}\right)^{2}$$

Square -\frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{7}{6}+\frac{121}{16}$$

Add -\frac{7}{6} to \frac{121}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{307}{48}$$

Factor x^{2}-\frac{11}{2}x+\frac{121}{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}.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{307}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{307}{48}}$$

Simplify.

$$x-\frac{11}{4}=\frac{\sqrt{921}}{12} x-\frac{11}{4}=-\frac{\sqrt{921}}{12}$$

Add \frac{11}{4} to both sides of the equation.

$$x=\frac{\sqrt{921}}{12}+\frac{11}{4} x=-\frac{\sqrt{921}}{12}+\frac{11}{4}$$