Solve 1/2x(1/2x-4)=33/4

$$\frac { 1 } { 2 } x ( \frac { 1 } { 2 } x - 4 ) = \frac { 33 } { 4 }$$

Solve for x

$$x=-3$$

$$x=11$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2}x by \frac{1}{2}x-4.

$$\left(\frac{1}{2}x\right)^{2}+\frac{1}{2}x\left(-4\right)=\frac{33}{4}$$

Expand \left(\frac{1}{2}x\right)^{2}.

$$\left(\frac{1}{2}\right)^{2}x^{2}+\frac{1}{2}x\left(-4\right)=\frac{33}{4}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1}{4}x^{2}+\frac{1}{2}x\left(-4\right)=\frac{33}{4}$$

Multiply \frac{1}{2} and -4 to get \frac{-4}{2}.

$$\frac{1}{4}x^{2}+\frac{-4}{2}x=\frac{33}{4}$$

Divide -4 by 2 to get -2.

$$\frac{1}{4}x^{2}-2x=\frac{33}{4}$$

Subtract \frac{33}{4} from both sides.

$$\frac{1}{4}x^{2}-2x-\frac{33}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{4} for a, -2 for b, and -\frac{33}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times \frac{1}{4}\left(-\frac{33}{4}\right)}}{2\times \frac{1}{4}}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times \frac{1}{4}\left(-\frac{33}{4}\right)}}{2\times \frac{1}{4}}$$

Multiply -4 times \frac{1}{4}.

$$x=\frac{-\left(-2\right)±\sqrt{4-\left(-\frac{33}{4}\right)}}{2\times \frac{1}{4}}$$

Multiply -1 times -\frac{33}{4}.

$$x=\frac{-\left(-2\right)±\sqrt{4+\frac{33}{4}}}{2\times \frac{1}{4}}$$

Add 4 to \frac{33}{4}.

$$x=\frac{-\left(-2\right)±\sqrt{\frac{49}{4}}}{2\times \frac{1}{4}}$$

Take the square root of \frac{49}{4}.

$$x=\frac{-\left(-2\right)±\frac{7}{2}}{2\times \frac{1}{4}}$$

The opposite of -2 is 2.

$$x=\frac{2±\frac{7}{2}}{2\times \frac{1}{4}}$$

Multiply 2 times \frac{1}{4}.

$$x=\frac{2±\frac{7}{2}}{\frac{1}{2}}$$

Now solve the equation x=\frac{2±\frac{7}{2}}{\frac{1}{2}} when ± is plus. Add 2 to \frac{7}{2}.

$$x=\frac{\frac{11}{2}}{\frac{1}{2}}$$

Divide \frac{11}{2} by \frac{1}{2} by multiplying \frac{11}{2} by the reciprocal of \frac{1}{2}.

$$x=11$$

Now solve the equation x=\frac{2±\frac{7}{2}}{\frac{1}{2}} when ± is minus. Subtract \frac{7}{2} from 2.

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

Divide -\frac{3}{2} by \frac{1}{2} by multiplying -\frac{3}{2} by the reciprocal of \frac{1}{2}.

$$x=-3$$

The equation is now solved.

$$x=11 x=-3$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2}x by \frac{1}{2}x-4.

$$\left(\frac{1}{2}x\right)^{2}+\frac{1}{2}x\left(-4\right)=\frac{33}{4}$$

Expand \left(\frac{1}{2}x\right)^{2}.

$$\left(\frac{1}{2}\right)^{2}x^{2}+\frac{1}{2}x\left(-4\right)=\frac{33}{4}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1}{4}x^{2}+\frac{1}{2}x\left(-4\right)=\frac{33}{4}$$

Multiply \frac{1}{2} and -4 to get \frac{-4}{2}.

$$\frac{1}{4}x^{2}+\frac{-4}{2}x=\frac{33}{4}$$

Divide -4 by 2 to get -2.

$$\frac{1}{4}x^{2}-2x=\frac{33}{4}$$

Multiply both sides by 4.

$$\frac{\frac{1}{4}x^{2}-2x}{\frac{1}{4}}=\frac{\frac{33}{4}}{\frac{1}{4}}$$

Dividing by \frac{1}{4} undoes the multiplication by \frac{1}{4}.

$$x^{2}+\left(-\frac{2}{\frac{1}{4}}\right)x=\frac{\frac{33}{4}}{\frac{1}{4}}$$

Divide -2 by \frac{1}{4} by multiplying -2 by the reciprocal of \frac{1}{4}.

$$x^{2}-8x=\frac{\frac{33}{4}}{\frac{1}{4}}$$

Divide \frac{33}{4} by \frac{1}{4} by multiplying \frac{33}{4} by the reciprocal of \frac{1}{4}.

$$x^{2}-8x=33$$

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

$$x^{2}-8x+\left(-4\right)^{2}=33+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=33+16$$

Add 33 to 16.

$$x^{2}-8x+16=49$$

Factor x^{2}-8x+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}.

$$\left(x-4\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{49}$$

Simplify.

$$x-4=7 x-4=-7$$

Add 4 to both sides of the equation.

$$x=11 x=-3$$