Solve -2a^2-2a-3=-4a^2

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

Solve for a

$$a = \frac{\sqrt{7} + 1}{2} \approx 1.822875656$$

$$a=\frac{1-\sqrt{7}}{2}\approx -0.822875656$$

Steps Using the Quadratic Formula

Add 4a^{2} to both sides.

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

Combine -2a^{2} and 4a^{2} to get 2a^{2}.

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

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

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

Square -2.

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

Multiply -4 times 2.

$$a=\frac{-\left(-2\right)±\sqrt{4-8\left(-3\right)}}{2\times 2}$$

Multiply -8 times -3.

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

Add 4 to 24.

$$a=\frac{-\left(-2\right)±\sqrt{28}}{2\times 2}$$

Take the square root of 28.

$$a=\frac{-\left(-2\right)±2\sqrt{7}}{2\times 2}$$

The opposite of -2 is 2.

$$a=\frac{2±2\sqrt{7}}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{2±2\sqrt{7}}{4}$$

Now solve the equation a=\frac{2±2\sqrt{7}}{4} when ± is plus. Add 2 to 2\sqrt{7}.

$$a=\frac{2\sqrt{7}+2}{4}$$

Divide 2+2\sqrt{7} by 4.

$$a=\frac{\sqrt{7}+1}{2}$$

Now solve the equation a=\frac{2±2\sqrt{7}}{4} when ± is minus. Subtract 2\sqrt{7} from 2.

$$a=\frac{2-2\sqrt{7}}{4}$$

Divide 2-2\sqrt{7} by 4.

$$a=\frac{1-\sqrt{7}}{2}$$

The equation is now solved.

$$a=\frac{\sqrt{7}+1}{2} a=\frac{1-\sqrt{7}}{2}$$

Steps for Completing the Square

Add 4a^{2} to both sides.

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

Combine -2a^{2} and 4a^{2} to get 2a^{2}.

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

Add 3 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide -2 by 2.

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

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add \frac{3}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-a+\frac{1}{4}=\frac{7}{4}$$

Factor a^{2}-a+\frac{1}{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}.

$$\left(a-\frac{1}{2}\right)^{2}=\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{1}{2}\right)^{2}}=\sqrt{\frac{7}{4}}$$

Simplify.

$$a-\frac{1}{2}=\frac{\sqrt{7}}{2} a-\frac{1}{2}=-\frac{\sqrt{7}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$a=\frac{\sqrt{7}+1}{2} a=\frac{1-\sqrt{7}}{2}$$