Solve 4-a=1/2a^2-5/2a+3

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

Solve for a

$$a = \frac{\sqrt{17} + 3}{2} \approx 3.561552813$$

$$a=\frac{3-\sqrt{17}}{2}\approx -0.561552813$$

Steps Using the Quadratic Formula

Subtract \frac{1}{2}a^{2} from both sides.

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

Add \frac{5}{2}a to both sides.

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

Combine -a and \frac{5}{2}a to get \frac{3}{2}a.

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

Subtract 3 from both sides.

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

Subtract 3 from 4 to get 1.

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

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.

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

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

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

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

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

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

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

Add \frac{9}{4} to 2.

$$a=\frac{-\frac{3}{2}±\sqrt{\frac{17}{4}}}{2\left(-\frac{1}{2}\right)}$$

Take the square root of \frac{17}{4}.

$$a=\frac{-\frac{3}{2}±\frac{\sqrt{17}}{2}}{2\left(-\frac{1}{2}\right)}$$

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

$$a=\frac{-\frac{3}{2}±\frac{\sqrt{17}}{2}}{-1}$$

Now solve the equation a=\frac{-\frac{3}{2}±\frac{\sqrt{17}}{2}}{-1} when ± is plus. Add -\frac{3}{2} to \frac{\sqrt{17}}{2}.

$$a=\frac{\sqrt{17}-3}{-2}$$

Divide \frac{-3+\sqrt{17}}{2} by -1.

$$a=\frac{3-\sqrt{17}}{2}$$

Now solve the equation a=\frac{-\frac{3}{2}±\frac{\sqrt{17}}{2}}{-1} when ± is minus. Subtract \frac{\sqrt{17}}{2} from -\frac{3}{2}.

$$a=\frac{-\sqrt{17}-3}{-2}$$

Divide \frac{-3-\sqrt{17}}{2} by -1.

$$a=\frac{\sqrt{17}+3}{2}$$

The equation is now solved.

$$a=\frac{3-\sqrt{17}}{2} a=\frac{\sqrt{17}+3}{2}$$

Steps for Completing the Square

Subtract \frac{1}{2}a^{2} from both sides.

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

Add \frac{5}{2}a to both sides.

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

Combine -a and \frac{5}{2}a to get \frac{3}{2}a.

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

Subtract 4 from both sides.

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

Subtract 4 from 3 to get -1.

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

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.

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

Multiply both sides by -2.

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

Dividing by -\frac{1}{2} undoes the multiplication by -\frac{1}{2}.

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

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

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

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

$$a^{2}-3a=2$$

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

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

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

$$a^{2}-3a+\frac{9}{4}=2+\frac{9}{4}$$

Add 2 to \frac{9}{4}.

$$a^{2}-3a+\frac{9}{4}=\frac{17}{4}$$

Factor a^{2}-3a+\frac{9}{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}.

$$\left(a-\frac{3}{2}\right)^{2}=\frac{17}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{3}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Simplify.

$$a-\frac{3}{2}=\frac{\sqrt{17}}{2} a-\frac{3}{2}=-\frac{\sqrt{17}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$a=\frac{\sqrt{17}+3}{2} a=\frac{3-\sqrt{17}}{2}$$