Solve 9-6a-3/2a+a^2=0

$$9 - 6 a - \frac { 3 } { 2 } a + a ^ { 2 } = 0$$

Solve for a

$$a = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$a=6$$

Steps Using the Quadratic Formula

Combine -6a and -\frac{3}{2}a to get -\frac{15}{2}a.

$$9-\frac{15}{2}a+a^{2}=0$$

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.

$$a^{2}-\frac{15}{2}a+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{15}{2} for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-\frac{15}{2}\right)±\sqrt{\left(-\frac{15}{2}\right)^{2}-4\times 9}}{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$a=\frac{-\left(-\frac{15}{2}\right)±\sqrt{\frac{225}{4}-4\times 9}}{2}$$

Multiply -4 times 9.

$$a=\frac{-\left(-\frac{15}{2}\right)±\sqrt{\frac{225}{4}-36}}{2}$$

Add \frac{225}{4} to -36.

$$a=\frac{-\left(-\frac{15}{2}\right)±\sqrt{\frac{81}{4}}}{2}$$

Take the square root of \frac{81}{4}.

$$a=\frac{-\left(-\frac{15}{2}\right)±\frac{9}{2}}{2}$$

The opposite of -\frac{15}{2} is \frac{15}{2}.

$$a=\frac{\frac{15}{2}±\frac{9}{2}}{2}$$

Now solve the equation a=\frac{\frac{15}{2}±\frac{9}{2}}{2} when ± is plus. Add \frac{15}{2} to \frac{9}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a=\frac{12}{2}$$

Divide 12 by 2.

$$a=6$$

Now solve the equation a=\frac{\frac{15}{2}±\frac{9}{2}}{2} when ± is minus. Subtract \frac{9}{2} from \frac{15}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

The equation is now solved.

$$a=6 a=\frac{3}{2}$$

Steps for Completing the Square

Combine -6a and -\frac{3}{2}a to get -\frac{15}{2}a.

$$9-\frac{15}{2}a+a^{2}=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{15}{2}a+a^{2}=-9$$

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.

$$a^{2}-\frac{15}{2}a=-9$$

Divide -\frac{15}{2}, the coefficient of the x term, by 2 to get -\frac{15}{4}. Then add the square of -\frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{15}{2}a+\left(-\frac{15}{4}\right)^{2}=-9+\left(-\frac{15}{4}\right)^{2}$$

Square -\frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{15}{2}a+\frac{225}{16}=-9+\frac{225}{16}$$

Add -9 to \frac{225}{16}.

$$a^{2}-\frac{15}{2}a+\frac{225}{16}=\frac{81}{16}$$

Factor a^{2}-\frac{15}{2}a+\frac{225}{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}.

$$\left(a-\frac{15}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{15}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$a-\frac{15}{4}=\frac{9}{4} a-\frac{15}{4}=-\frac{9}{4}$$

Add \frac{15}{4} to both sides of the equation.

$$a=6 a=\frac{3}{2}$$