Solve -4/3a+4=a^2-2a+1-4

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

Solve for a

$$a = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$a=3$$

Steps Using the Quadratic Formula

Subtract 4 from 1 to get -3.

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

Subtract a^{2} from both sides.

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

Add 2a to both sides.

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

Combine -\frac{4}{3}a and 2a to get \frac{2}{3}a.

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

Add 3 to both sides.

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

Add 4 and 3 to get 7.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{2}{3} for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\frac{2}{3}±\sqrt{\left(\frac{2}{3}\right)^{2}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$a=\frac{-\frac{2}{3}±\sqrt{\frac{4}{9}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-\frac{2}{3}±\sqrt{\frac{4}{9}+4\times 7}}{2\left(-1\right)}$$

Multiply 4 times 7.

$$a=\frac{-\frac{2}{3}±\sqrt{\frac{4}{9}+28}}{2\left(-1\right)}$$

Add \frac{4}{9} to 28.

$$a=\frac{-\frac{2}{3}±\sqrt{\frac{256}{9}}}{2\left(-1\right)}$$

Take the square root of \frac{256}{9}.

$$a=\frac{-\frac{2}{3}±\frac{16}{3}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-\frac{2}{3}±\frac{16}{3}}{-2}$$

Now solve the equation a=\frac{-\frac{2}{3}±\frac{16}{3}}{-2} when ± is plus. Add -\frac{2}{3} to \frac{16}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide \frac{14}{3} by -2.

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

Now solve the equation a=\frac{-\frac{2}{3}±\frac{16}{3}}{-2} when ± is minus. Subtract \frac{16}{3} from -\frac{2}{3} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$a=-\frac{6}{-2}$$

Divide -6 by -2.

$$a=3$$

The equation is now solved.

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

Steps for Completing the Square

Subtract 4 from 1 to get -3.

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

Subtract a^{2} from both sides.

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

Add 2a to both sides.

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

Combine -\frac{4}{3}a and 2a to get \frac{2}{3}a.

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

Subtract 4 from both sides.

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

Subtract 4 from -3 to get -7.

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

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.

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

Divide both sides by -1.

$$\frac{-a^{2}+\frac{2}{3}a}{-1}=-\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\frac{\frac{2}{3}}{-1}a=-\frac{7}{-1}$$

Divide \frac{2}{3} by -1.

$$a^{2}-\frac{2}{3}a=-\frac{7}{-1}$$

Divide -7 by -1.

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

Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{2}{3}a+\left(-\frac{1}{3}\right)^{2}=7+\left(-\frac{1}{3}\right)^{2}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{2}{3}a+\frac{1}{9}=7+\frac{1}{9}$$

Add 7 to \frac{1}{9}.

$$a^{2}-\frac{2}{3}a+\frac{1}{9}=\frac{64}{9}$$

Factor a^{2}-\frac{2}{3}a+\frac{1}{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}.

$$\left(a-\frac{1}{3}\right)^{2}=\frac{64}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{1}{3}\right)^{2}}=\sqrt{\frac{64}{9}}$$

Simplify.

$$a-\frac{1}{3}=\frac{8}{3} a-\frac{1}{3}=-\frac{8}{3}$$

Add \frac{1}{3} to both sides of the equation.

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