Solve 2a^2+2-4a+a^2-a-2=16

$$2 a ^ { 2 } + 2 - 4 a + a ^ { 2 } - a - 2 = 16$$

Solve for a

$$a = \frac{\sqrt{217} + 5}{6} \approx 3.288486644$$

$$a=\frac{5-\sqrt{217}}{6}\approx -1.621819977$$

Steps Using the Quadratic Formula

Combine 2a^{2} and a^{2} to get 3a^{2}.

$$3a^{2}+2-4a-a-2=16$$

Combine -4a and -a to get -5a.

$$3a^{2}+2-5a-2=16$$

Subtract 2 from 2 to get 0.

$$3a^{2}-5a=16$$

Subtract 16 from both sides.

$$3a^{2}-5a-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -5 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-16\right)}}{2\times 3}$$

Square -5.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\times 3\left(-16\right)}}{2\times 3}$$

Multiply -4 times 3.

$$a=\frac{-\left(-5\right)±\sqrt{25-12\left(-16\right)}}{2\times 3}$$

Multiply -12 times -16.

$$a=\frac{-\left(-5\right)±\sqrt{25+192}}{2\times 3}$$

Add 25 to 192.

$$a=\frac{-\left(-5\right)±\sqrt{217}}{2\times 3}$$

The opposite of -5 is 5.

$$a=\frac{5±\sqrt{217}}{2\times 3}$$

Multiply 2 times 3.

$$a=\frac{5±\sqrt{217}}{6}$$

Now solve the equation a=\frac{5±\sqrt{217}}{6} when ± is plus. Add 5 to \sqrt{217}.

$$a=\frac{\sqrt{217}+5}{6}$$

Now solve the equation a=\frac{5±\sqrt{217}}{6} when ± is minus. Subtract \sqrt{217} from 5.

$$a=\frac{5-\sqrt{217}}{6}$$

The equation is now solved.

$$a=\frac{\sqrt{217}+5}{6} a=\frac{5-\sqrt{217}}{6}$$

Steps for Completing the Square

Combine 2a^{2} and a^{2} to get 3a^{2}.

$$3a^{2}+2-4a-a-2=16$$

Combine -4a and -a to get -5a.

$$3a^{2}+2-5a-2=16$$

Subtract 2 from 2 to get 0.

$$3a^{2}-5a=16$$

Divide both sides by 3.

$$\frac{3a^{2}-5a}{3}=\frac{16}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$a^{2}-\frac{5}{3}a=\frac{16}{3}$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{5}{3}a+\left(-\frac{5}{6}\right)^{2}=\frac{16}{3}+\left(-\frac{5}{6}\right)^{2}$$

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{5}{3}a+\frac{25}{36}=\frac{16}{3}+\frac{25}{36}$$

Add \frac{16}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{5}{3}a+\frac{25}{36}=\frac{217}{36}$$

Factor a^{2}-\frac{5}{3}a+\frac{25}{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}.

$$\left(a-\frac{5}{6}\right)^{2}=\frac{217}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{5}{6}\right)^{2}}=\sqrt{\frac{217}{36}}$$

Simplify.

$$a-\frac{5}{6}=\frac{\sqrt{217}}{6} a-\frac{5}{6}=-\frac{\sqrt{217}}{6}$$

Add \frac{5}{6} to both sides of the equation.

$$a=\frac{\sqrt{217}+5}{6} a=\frac{5-\sqrt{217}}{6}$$