Solve .4a^2+4a/4=80/4

$$.4 a ^ { 2 } + \frac { 4 a } { 4 } = \frac { 80 } { 4 }$$

Solve for a

$$a = \frac{5 \sqrt{33} - 5}{4} \approx 5.930703308$$

$$a=\frac{-5\sqrt{33}-5}{4}\approx -8.430703308$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4.

$$1.6a^{2}+4a=80$$

Subtract 80 from both sides.

$$1.6a^{2}+4a-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.6 for a, 4 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-4±\sqrt{4^{2}-4\times 1.6\left(-80\right)}}{2\times 1.6}$$

Square 4.

$$a=\frac{-4±\sqrt{16-4\times 1.6\left(-80\right)}}{2\times 1.6}$$

Multiply -4 times 1.6.

$$a=\frac{-4±\sqrt{16-6.4\left(-80\right)}}{2\times 1.6}$$

Multiply -6.4 times -80.

$$a=\frac{-4±\sqrt{16+512}}{2\times 1.6}$$

Add 16 to 512.

$$a=\frac{-4±\sqrt{528}}{2\times 1.6}$$

Take the square root of 528.

$$a=\frac{-4±4\sqrt{33}}{2\times 1.6}$$

Multiply 2 times 1.6.

$$a=\frac{-4±4\sqrt{33}}{3.2}$$

Now solve the equation a=\frac{-4±4\sqrt{33}}{3.2} when ± is plus. Add -4 to 4\sqrt{33}.

$$a=\frac{4\sqrt{33}-4}{3.2}$$

Divide -4+4\sqrt{33} by 3.2 by multiplying -4+4\sqrt{33} by the reciprocal of 3.2.

$$a=\frac{5\sqrt{33}-5}{4}$$

Now solve the equation a=\frac{-4±4\sqrt{33}}{3.2} when ± is minus. Subtract 4\sqrt{33} from -4.

$$a=\frac{-4\sqrt{33}-4}{3.2}$$

Divide -4-4\sqrt{33} by 3.2 by multiplying -4-4\sqrt{33} by the reciprocal of 3.2.

$$a=\frac{-5\sqrt{33}-5}{4}$$

The equation is now solved.

$$a=\frac{5\sqrt{33}-5}{4} a=\frac{-5\sqrt{33}-5}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 4.

$$1.6a^{2}+4a=80$$

Divide both sides of the equation by 1.6, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.6a^{2}+4a}{1.6}=\frac{80}{1.6}$$

Dividing by 1.6 undoes the multiplication by 1.6.

$$a^{2}+\frac{4}{1.6}a=\frac{80}{1.6}$$

Divide 4 by 1.6 by multiplying 4 by the reciprocal of 1.6.

$$a^{2}+2.5a=\frac{80}{1.6}$$

Divide 80 by 1.6 by multiplying 80 by the reciprocal of 1.6.

$$a^{2}+2.5a=50$$

Divide 2.5, the coefficient of the x term, by 2 to get 1.25. Then add the square of 1.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+2.5a+1.25^{2}=50+1.25^{2}$$

Square 1.25 by squaring both the numerator and the denominator of the fraction.

$$a^{2}+2.5a+1.5625=50+1.5625$$

Add 50 to 1.5625.

$$a^{2}+2.5a+1.5625=51.5625$$

Factor a^{2}+2.5a+1.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+1.25\right)^{2}=51.5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+1.25\right)^{2}}=\sqrt{51.5625}$$

Simplify.

$$a+1.25=\frac{5\sqrt{33}}{4} a+1.25=-\frac{5\sqrt{33}}{4}$$

Subtract 1.25 from both sides of the equation.

$$a=\frac{5\sqrt{33}-5}{4} a=\frac{-5\sqrt{33}-5}{4}$$