Solve 3/4a^2-3a+4-5=2(a-4)

$$\frac { 3 } { 4 } a ^ { 2 } - 3 a + 4 - 5 = 2 ( a - 4 )$$

Solve for a

$$a=2$$

$$a = \frac{14}{3} = 4\frac{2}{3} \approx 4.666666667$$

Steps Using the Quadratic Formula

Subtract 5 from 4 to get -1.

$$\frac{3}{4}a^{2}-3a-1=2\left(a-4\right)$$

Use the distributive property to multiply 2 by a-4.

$$\frac{3}{4}a^{2}-3a-1=2a-8$$

Subtract 2a from both sides.

$$\frac{3}{4}a^{2}-3a-1-2a=-8$$

Combine -3a and -2a to get -5a.

$$\frac{3}{4}a^{2}-5a-1=-8$$

Add 8 to both sides.

$$\frac{3}{4}a^{2}-5a-1+8=0$$

Add -1 and 8 to get 7.

$$\frac{3}{4}a^{2}-5a+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{4} for a, -5 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times \frac{3}{4}\times 7}}{2\times \frac{3}{4}}$$

Square -5.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\times \frac{3}{4}\times 7}}{2\times \frac{3}{4}}$$

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

$$a=\frac{-\left(-5\right)±\sqrt{25-3\times 7}}{2\times \frac{3}{4}}$$

Multiply -3 times 7.

$$a=\frac{-\left(-5\right)±\sqrt{25-21}}{2\times \frac{3}{4}}$$

Add 25 to -21.

$$a=\frac{-\left(-5\right)±\sqrt{4}}{2\times \frac{3}{4}}$$

Take the square root of 4.

$$a=\frac{-\left(-5\right)±2}{2\times \frac{3}{4}}$$

The opposite of -5 is 5.

$$a=\frac{5±2}{2\times \frac{3}{4}}$$

Multiply 2 times \frac{3}{4}.

$$a=\frac{5±2}{\frac{3}{2}}$$

Now solve the equation a=\frac{5±2}{\frac{3}{2}} when ± is plus. Add 5 to 2.

$$a=\frac{7}{\frac{3}{2}}$$

Divide 7 by \frac{3}{2} by multiplying 7 by the reciprocal of \frac{3}{2}.

$$a=\frac{14}{3}$$

Now solve the equation a=\frac{5±2}{\frac{3}{2}} when ± is minus. Subtract 2 from 5.

$$a=\frac{3}{\frac{3}{2}}$$

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

$$a=2$$

The equation is now solved.

$$a=\frac{14}{3} a=2$$

Steps for Completing the Square

Subtract 5 from 4 to get -1.

$$\frac{3}{4}a^{2}-3a-1=2\left(a-4\right)$$

Use the distributive property to multiply 2 by a-4.

$$\frac{3}{4}a^{2}-3a-1=2a-8$$

Subtract 2a from both sides.

$$\frac{3}{4}a^{2}-3a-1-2a=-8$$

Combine -3a and -2a to get -5a.

$$\frac{3}{4}a^{2}-5a-1=-8$$

Add 1 to both sides.

$$\frac{3}{4}a^{2}-5a=-8+1$$

Add -8 and 1 to get -7.

$$\frac{3}{4}a^{2}-5a=-7$$

Divide both sides of the equation by \frac{3}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{3}{4}a^{2}-5a}{\frac{3}{4}}=-\frac{7}{\frac{3}{4}}$$

Dividing by \frac{3}{4} undoes the multiplication by \frac{3}{4}.

$$a^{2}+\left(-\frac{5}{\frac{3}{4}}\right)a=-\frac{7}{\frac{3}{4}}$$

Divide -5 by \frac{3}{4} by multiplying -5 by the reciprocal of \frac{3}{4}.

$$a^{2}-\frac{20}{3}a=-\frac{7}{\frac{3}{4}}$$

Divide -7 by \frac{3}{4} by multiplying -7 by the reciprocal of \frac{3}{4}.

$$a^{2}-\frac{20}{3}a=-\frac{28}{3}$$

Divide -\frac{20}{3}, the coefficient of the x term, by 2 to get -\frac{10}{3}. Then add the square of -\frac{10}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{20}{3}a+\left(-\frac{10}{3}\right)^{2}=-\frac{28}{3}+\left(-\frac{10}{3}\right)^{2}$$

Square -\frac{10}{3} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{20}{3}a+\frac{100}{9}=-\frac{28}{3}+\frac{100}{9}$$

Add -\frac{28}{3} to \frac{100}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{20}{3}a+\frac{100}{9}=\frac{16}{9}$$

Factor a^{2}-\frac{20}{3}a+\frac{100}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{10}{3}\right)^{2}=\frac{16}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{10}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Simplify.

$$a-\frac{10}{3}=\frac{4}{3} a-\frac{10}{3}=-\frac{4}{3}$$

Add \frac{10}{3} to both sides of the equation.

$$a=\frac{14}{3} a=2$$