Solve 0=a^2+5a-40

$$0 = a ^ { 2 } + 5 a - 40$$

Solve for a

$$a = \frac{\sqrt{185} - 5}{2} \approx 4.300735254$$

$$a=\frac{-\sqrt{185}-5}{2}\approx -9.300735254$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$a^{2}+5a-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-5±\sqrt{5^{2}-4\left(-40\right)}}{2}$$

Square 5.

$$a=\frac{-5±\sqrt{25-4\left(-40\right)}}{2}$$

Multiply -4 times -40.

$$a=\frac{-5±\sqrt{25+160}}{2}$$

Add 25 to 160.

$$a=\frac{-5±\sqrt{185}}{2}$$

Now solve the equation a=\frac{-5±\sqrt{185}}{2} when ± is plus. Add -5 to \sqrt{185}.

$$a=\frac{\sqrt{185}-5}{2}$$

Now solve the equation a=\frac{-5±\sqrt{185}}{2} when ± is minus. Subtract \sqrt{185} from -5.

$$a=\frac{-\sqrt{185}-5}{2}$$

The equation is now solved.

$$a=\frac{\sqrt{185}-5}{2} a=\frac{-\sqrt{185}-5}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$a^{2}+5a-40=0$$

Add 40 to both sides. Anything plus zero gives itself.

$$a^{2}+5a=40$$

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

$$a^{2}+5a+\left(\frac{5}{2}\right)^{2}=40+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+5a+\frac{25}{4}=40+\frac{25}{4}$$

Add 40 to \frac{25}{4}.

$$a^{2}+5a+\frac{25}{4}=\frac{185}{4}$$

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

$$\left(a+\frac{5}{2}\right)^{2}=\frac{185}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{5}{2}\right)^{2}}=\sqrt{\frac{185}{4}}$$

Simplify.

$$a+\frac{5}{2}=\frac{\sqrt{185}}{2} a+\frac{5}{2}=-\frac{\sqrt{185}}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$a=\frac{\sqrt{185}-5}{2} a=\frac{-\sqrt{185}-5}{2}$$