Solve 8-2a-32+8a^2-40a/5=5/2

$$8-2a- \frac{ 32+8 { a }^{ 2 } -40a }{ 5 } = \frac{ 5 }{ 2 }$$

Solve for a

$$a = \frac{3 \sqrt{21} + 15}{8} \approx 3.593465886$$

$$a=\frac{15-3\sqrt{21}}{8}\approx 0.156534114$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10, the least common multiple of 5,2.

$$80-20a-2\left(32+8a^{2}-40a\right)=25$$

Use the distributive property to multiply -2 by 32+8a^{2}-40a.

$$80-20a-64-16a^{2}+80a=25$$

Subtract 64 from 80 to get 16.

$$16-20a-16a^{2}+80a=25$$

Combine -20a and 80a to get 60a.

$$16+60a-16a^{2}=25$$

Subtract 25 from both sides.

$$16+60a-16a^{2}-25=0$$

Subtract 25 from 16 to get -9.

$$-9+60a-16a^{2}=0$$

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.

$$-16a^{2}+60a-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 60 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-60±\sqrt{60^{2}-4\left(-16\right)\left(-9\right)}}{2\left(-16\right)}$$

Square 60.

$$a=\frac{-60±\sqrt{3600-4\left(-16\right)\left(-9\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$a=\frac{-60±\sqrt{3600+64\left(-9\right)}}{2\left(-16\right)}$$

Multiply 64 times -9.

$$a=\frac{-60±\sqrt{3600-576}}{2\left(-16\right)}$$

Add 3600 to -576.

$$a=\frac{-60±\sqrt{3024}}{2\left(-16\right)}$$

Take the square root of 3024.

$$a=\frac{-60±12\sqrt{21}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$a=\frac{-60±12\sqrt{21}}{-32}$$

Now solve the equation a=\frac{-60±12\sqrt{21}}{-32} when ± is plus. Add -60 to 12\sqrt{21}.

$$a=\frac{12\sqrt{21}-60}{-32}$$

Divide -60+12\sqrt{21} by -32.

$$a=\frac{15-3\sqrt{21}}{8}$$

Now solve the equation a=\frac{-60±12\sqrt{21}}{-32} when ± is minus. Subtract 12\sqrt{21} from -60.

$$a=\frac{-12\sqrt{21}-60}{-32}$$

Divide -60-12\sqrt{21} by -32.

$$a=\frac{3\sqrt{21}+15}{8}$$

The equation is now solved.

$$a=\frac{15-3\sqrt{21}}{8} a=\frac{3\sqrt{21}+15}{8}$$

Steps for Completing the Square

Multiply both sides of the equation by 10, the least common multiple of 5,2.

$$80-20a-2\left(32+8a^{2}-40a\right)=25$$

Use the distributive property to multiply -2 by 32+8a^{2}-40a.

$$80-20a-64-16a^{2}+80a=25$$

Subtract 64 from 80 to get 16.

$$16-20a-16a^{2}+80a=25$$

Combine -20a and 80a to get 60a.

$$16+60a-16a^{2}=25$$

Subtract 16 from both sides.

$$60a-16a^{2}=25-16$$

Subtract 16 from 25 to get 9.

$$60a-16a^{2}=9$$

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.

$$-16a^{2}+60a=9$$

Divide both sides by -16.

$$\frac{-16a^{2}+60a}{-16}=\frac{9}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$a^{2}+\frac{60}{-16}a=\frac{9}{-16}$$

Reduce the fraction \frac{60}{-16} to lowest terms by extracting and canceling out 4.

$$a^{2}-\frac{15}{4}a=\frac{9}{-16}$$

Divide 9 by -16.

$$a^{2}-\frac{15}{4}a=-\frac{9}{16}$$

Divide -\frac{15}{4}, the coefficient of the x term, by 2 to get -\frac{15}{8}. Then add the square of -\frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{15}{4}a+\left(-\frac{15}{8}\right)^{2}=-\frac{9}{16}+\left(-\frac{15}{8}\right)^{2}$$

Square -\frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{15}{4}a+\frac{225}{64}=-\frac{9}{16}+\frac{225}{64}$$

Add -\frac{9}{16} to \frac{225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{15}{4}a+\frac{225}{64}=\frac{189}{64}$$

Factor a^{2}-\frac{15}{4}a+\frac{225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{15}{8}\right)^{2}=\frac{189}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{15}{8}\right)^{2}}=\sqrt{\frac{189}{64}}$$

Simplify.

$$a-\frac{15}{8}=\frac{3\sqrt{21}}{8} a-\frac{15}{8}=-\frac{3\sqrt{21}}{8}$$

Add \frac{15}{8} to both sides of the equation.

$$a=\frac{3\sqrt{21}+15}{8} a=\frac{15-3\sqrt{21}}{8}$$