Solve 1/2(8-2t)*3/16t=3

$$\frac { 1 } { 2 } ( 8 - 2 t ) \times \frac { 3 } { 16 } t = 3$$

Solve for t

$$t=2+2\sqrt{3}i\approx 2+3.464101615i$$

$$t=-2\sqrt{3}i+2\approx 2-3.464101615i$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} times \frac{3}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 3}{2\times 16}\left(8-2t\right)t=3$$

Do the multiplications in the fraction \frac{1\times 3}{2\times 16}.

$$\frac{3}{32}\left(8-2t\right)t=3$$

Use the distributive property to multiply \frac{3}{32} by 8-2t.

$$\left(\frac{3}{32}\times 8+\frac{3}{32}\left(-2\right)t\right)t=3$$

Express \frac{3}{32}\times 8 as a single fraction.

$$\left(\frac{3\times 8}{32}+\frac{3}{32}\left(-2\right)t\right)t=3$$

Multiply 3 and 8 to get 24.

$$\left(\frac{24}{32}+\frac{3}{32}\left(-2\right)t\right)t=3$$

Reduce the fraction \frac{24}{32} to lowest terms by extracting and canceling out 8.

$$\left(\frac{3}{4}+\frac{3}{32}\left(-2\right)t\right)t=3$$

Express \frac{3}{32}\left(-2\right) as a single fraction.

$$\left(\frac{3}{4}+\frac{3\left(-2\right)}{32}t\right)t=3$$

Multiply 3 and -2 to get -6.

$$\left(\frac{3}{4}+\frac{-6}{32}t\right)t=3$$

Reduce the fraction \frac{-6}{32} to lowest terms by extracting and canceling out 2.

$$\left(\frac{3}{4}-\frac{3}{16}t\right)t=3$$

Use the distributive property to multiply \frac{3}{4}-\frac{3}{16}t by t.

$$\frac{3}{4}t-\frac{3}{16}tt=3$$

Multiply t and t to get t^{2}.

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

Subtract 3 from both sides.

$$\frac{3}{4}t-\frac{3}{16}t^{2}-3=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-\frac{3}{16}t^{2}+\frac{3}{4}t-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{3}{16} for a, \frac{3}{4} for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\frac{3}{4}±\sqrt{\left(\frac{3}{4}\right)^{2}-4\left(-\frac{3}{16}\right)\left(-3\right)}}{2\left(-\frac{3}{16}\right)}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\frac{3}{4}±\sqrt{\frac{9}{16}-4\left(-\frac{3}{16}\right)\left(-3\right)}}{2\left(-\frac{3}{16}\right)}$$

Multiply -4 times -\frac{3}{16}.

$$t=\frac{-\frac{3}{4}±\sqrt{\frac{9}{16}+\frac{3}{4}\left(-3\right)}}{2\left(-\frac{3}{16}\right)}$$

Multiply \frac{3}{4} times -3.

$$t=\frac{-\frac{3}{4}±\sqrt{\frac{9}{16}-\frac{9}{4}}}{2\left(-\frac{3}{16}\right)}$$

Add \frac{9}{16} to -\frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\frac{3}{4}±\sqrt{-\frac{27}{16}}}{2\left(-\frac{3}{16}\right)}$$

Take the square root of -\frac{27}{16}.

$$t=\frac{-\frac{3}{4}±\frac{3\sqrt{3}i}{4}}{2\left(-\frac{3}{16}\right)}$$

Multiply 2 times -\frac{3}{16}.

$$t=\frac{-\frac{3}{4}±\frac{3\sqrt{3}i}{4}}{-\frac{3}{8}}$$

Now solve the equation t=\frac{-\frac{3}{4}±\frac{3\sqrt{3}i}{4}}{-\frac{3}{8}} when ± is plus. Add -\frac{3}{4} to \frac{3i\sqrt{3}}{4}.

$$t=\frac{-3+3\sqrt{3}i}{-\frac{3}{8}\times 4}$$

Divide \frac{-3+3i\sqrt{3}}{4} by -\frac{3}{8} by multiplying \frac{-3+3i\sqrt{3}}{4} by the reciprocal of -\frac{3}{8}.

$$t=-2\sqrt{3}i+2$$

Now solve the equation t=\frac{-\frac{3}{4}±\frac{3\sqrt{3}i}{4}}{-\frac{3}{8}} when ± is minus. Subtract \frac{3i\sqrt{3}}{4} from -\frac{3}{4}.

$$t=\frac{-3\sqrt{3}i-3}{-\frac{3}{8}\times 4}$$

Divide \frac{-3-3i\sqrt{3}}{4} by -\frac{3}{8} by multiplying \frac{-3-3i\sqrt{3}}{4} by the reciprocal of -\frac{3}{8}.

$$t=2+2\sqrt{3}i$$

The equation is now solved.

$$t=-2\sqrt{3}i+2 t=2+2\sqrt{3}i$$

Steps for Completing the Square

Multiply \frac{1}{2} times \frac{3}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 3}{2\times 16}\left(8-2t\right)t=3$$

Do the multiplications in the fraction \frac{1\times 3}{2\times 16}.

$$\frac{3}{32}\left(8-2t\right)t=3$$

Use the distributive property to multiply \frac{3}{32} by 8-2t.

$$\left(\frac{3}{32}\times 8+\frac{3}{32}\left(-2\right)t\right)t=3$$

Express \frac{3}{32}\times 8 as a single fraction.

$$\left(\frac{3\times 8}{32}+\frac{3}{32}\left(-2\right)t\right)t=3$$

Multiply 3 and 8 to get 24.

$$\left(\frac{24}{32}+\frac{3}{32}\left(-2\right)t\right)t=3$$

Reduce the fraction \frac{24}{32} to lowest terms by extracting and canceling out 8.

$$\left(\frac{3}{4}+\frac{3}{32}\left(-2\right)t\right)t=3$$

Express \frac{3}{32}\left(-2\right) as a single fraction.

$$\left(\frac{3}{4}+\frac{3\left(-2\right)}{32}t\right)t=3$$

Multiply 3 and -2 to get -6.

$$\left(\frac{3}{4}+\frac{-6}{32}t\right)t=3$$

Reduce the fraction \frac{-6}{32} to lowest terms by extracting and canceling out 2.

$$\left(\frac{3}{4}-\frac{3}{16}t\right)t=3$$

Use the distributive property to multiply \frac{3}{4}-\frac{3}{16}t by t.

$$\frac{3}{4}t-\frac{3}{16}tt=3$$

Multiply t and t to get t^{2}.

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides of the equation by -\frac{3}{16}, which is the same as multiplying both sides by the reciprocal of the fraction.

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

Dividing by -\frac{3}{16} undoes the multiplication by -\frac{3}{16}.

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

Divide \frac{3}{4} by -\frac{3}{16} by multiplying \frac{3}{4} by the reciprocal of -\frac{3}{16}.

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

Divide 3 by -\frac{3}{16} by multiplying 3 by the reciprocal of -\frac{3}{16}.

$$t^{2}-4t=-16$$

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

$$t^{2}-4t+\left(-2\right)^{2}=-16+\left(-2\right)^{2}$$

Square -2.

$$t^{2}-4t+4=-16+4$$

Add -16 to 4.

$$t^{2}-4t+4=-12$$

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

$$\left(t-2\right)^{2}=-12$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-2\right)^{2}}=\sqrt{-12}$$

Simplify.

$$t-2=2\sqrt{3}i t-2=-2\sqrt{3}i$$

Add 2 to both sides of the equation.

$$t=2+2\sqrt{3}i t=-2\sqrt{3}i+2$$