Solve 5a-a^2/4=2a^2-4a-6

$$\frac { 5 a - a ^ { 2 } } { 4 } = 2 a ^ { 2 } - 4 a - 6$$

Solve for a

$$a = \frac{\sqrt{145} + 7}{6} \approx 3.173599096$$

$$a=\frac{7-\sqrt{145}}{6}\approx -0.840265763$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4.

$$5a-a^{2}=8a^{2}-16a-24$$

Subtract 8a^{2} from both sides.

$$5a-a^{2}-8a^{2}=-16a-24$$

Combine -a^{2} and -8a^{2} to get -9a^{2}.

$$5a-9a^{2}=-16a-24$$

Add 16a to both sides.

$$5a-9a^{2}+16a=-24$$

Combine 5a and 16a to get 21a.

$$21a-9a^{2}=-24$$

Add 24 to both sides.

$$21a-9a^{2}+24=0$$

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.

$$-9a^{2}+21a+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 21 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-21±\sqrt{21^{2}-4\left(-9\right)\times 24}}{2\left(-9\right)}$$

Square 21.

$$a=\frac{-21±\sqrt{441-4\left(-9\right)\times 24}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$a=\frac{-21±\sqrt{441+36\times 24}}{2\left(-9\right)}$$

Multiply 36 times 24.

$$a=\frac{-21±\sqrt{441+864}}{2\left(-9\right)}$$

Add 441 to 864.

$$a=\frac{-21±\sqrt{1305}}{2\left(-9\right)}$$

Take the square root of 1305.

$$a=\frac{-21±3\sqrt{145}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$a=\frac{-21±3\sqrt{145}}{-18}$$

Now solve the equation a=\frac{-21±3\sqrt{145}}{-18} when ± is plus. Add -21 to 3\sqrt{145}.

$$a=\frac{3\sqrt{145}-21}{-18}$$

Divide -21+3\sqrt{145} by -18.

$$a=\frac{7-\sqrt{145}}{6}$$

Now solve the equation a=\frac{-21±3\sqrt{145}}{-18} when ± is minus. Subtract 3\sqrt{145} from -21.

$$a=\frac{-3\sqrt{145}-21}{-18}$$

Divide -21-3\sqrt{145} by -18.

$$a=\frac{\sqrt{145}+7}{6}$$

The equation is now solved.

$$a=\frac{7-\sqrt{145}}{6} a=\frac{\sqrt{145}+7}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 4.

$$5a-a^{2}=8a^{2}-16a-24$$

Subtract 8a^{2} from both sides.

$$5a-a^{2}-8a^{2}=-16a-24$$

Combine -a^{2} and -8a^{2} to get -9a^{2}.

$$5a-9a^{2}=-16a-24$$

Add 16a to both sides.

$$5a-9a^{2}+16a=-24$$

Combine 5a and 16a to get 21a.

$$21a-9a^{2}=-24$$

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.

$$-9a^{2}+21a=-24$$

Divide both sides by -9.

$$\frac{-9a^{2}+21a}{-9}=-\frac{24}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$a^{2}+\frac{21}{-9}a=-\frac{24}{-9}$$

Reduce the fraction \frac{21}{-9} to lowest terms by extracting and canceling out 3.

$$a^{2}-\frac{7}{3}a=-\frac{24}{-9}$$

Reduce the fraction \frac{-24}{-9} to lowest terms by extracting and canceling out 3.

$$a^{2}-\frac{7}{3}a=\frac{8}{3}$$

Divide -\frac{7}{3}, the coefficient of the x term, by 2 to get -\frac{7}{6}. Then add the square of -\frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{7}{3}a+\left(-\frac{7}{6}\right)^{2}=\frac{8}{3}+\left(-\frac{7}{6}\right)^{2}$$

Square -\frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{7}{3}a+\frac{49}{36}=\frac{8}{3}+\frac{49}{36}$$

Add \frac{8}{3} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{7}{3}a+\frac{49}{36}=\frac{145}{36}$$

Factor a^{2}-\frac{7}{3}a+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{7}{6}\right)^{2}=\frac{145}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{7}{6}\right)^{2}}=\sqrt{\frac{145}{36}}$$

Simplify.

$$a-\frac{7}{6}=\frac{\sqrt{145}}{6} a-\frac{7}{6}=-\frac{\sqrt{145}}{6}$$

Add \frac{7}{6} to both sides of the equation.

$$a=\frac{\sqrt{145}+7}{6} a=\frac{7-\sqrt{145}}{6}$$