Solve a^2+15a-2500=0quad8a^2

$$a ^ { 2 } + 15 a - 2500 = 0 \quad 8 a ^ { 2 }$$

Solve for a

$$a = \frac{5 \sqrt{409} - 15}{2} \approx 43.05937104$$

$$a=\frac{-5\sqrt{409}-15}{2}\approx -58.05937104$$

Steps Using the Quadratic Formula

Multiply 0 and 8 to get 0.

$$a^{2}+15a-2500=0a^{2}$$

Anything times zero gives zero.

$$a^{2}+15a-2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 15 for b, and -2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-15±\sqrt{15^{2}-4\left(-2500\right)}}{2}$$

Square 15.

$$a=\frac{-15±\sqrt{225-4\left(-2500\right)}}{2}$$

Multiply -4 times -2500.

$$a=\frac{-15±\sqrt{225+10000}}{2}$$

Add 225 to 10000.

$$a=\frac{-15±\sqrt{10225}}{2}$$

Take the square root of 10225.

$$a=\frac{-15±5\sqrt{409}}{2}$$

Now solve the equation a=\frac{-15±5\sqrt{409}}{2} when ± is plus. Add -15 to 5\sqrt{409}.

$$a=\frac{5\sqrt{409}-15}{2}$$

Now solve the equation a=\frac{-15±5\sqrt{409}}{2} when ± is minus. Subtract 5\sqrt{409} from -15.

$$a=\frac{-5\sqrt{409}-15}{2}$$

The equation is now solved.

$$a=\frac{5\sqrt{409}-15}{2} a=\frac{-5\sqrt{409}-15}{2}$$

Steps for Completing the Square

Multiply 0 and 8 to get 0.

$$a^{2}+15a-2500=0a^{2}$$

Anything times zero gives zero.

$$a^{2}+15a-2500=0$$

Add 2500 to both sides. Anything plus zero gives itself.

$$a^{2}+15a=2500$$

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

$$a^{2}+15a+\left(\frac{15}{2}\right)^{2}=2500+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+15a+\frac{225}{4}=2500+\frac{225}{4}$$

Add 2500 to \frac{225}{4}.

$$a^{2}+15a+\frac{225}{4}=\frac{10225}{4}$$

Factor a^{2}+15a+\frac{225}{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}.

$$\left(a+\frac{15}{2}\right)^{2}=\frac{10225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{15}{2}\right)^{2}}=\sqrt{\frac{10225}{4}}$$

Simplify.

$$a+\frac{15}{2}=\frac{5\sqrt{409}}{2} a+\frac{15}{2}=-\frac{5\sqrt{409}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$a=\frac{5\sqrt{409}-15}{2} a=\frac{-5\sqrt{409}-15}{2}$$