Solve -390a-(3+10a)a=6

$$- 390 a - ( 3 + 10 a ) a = 6$$

Solve for a

$$a=\frac{\sqrt{154209}-393}{20}\approx -0.015273111$$

$$a=\frac{-\sqrt{154209}-393}{20}\approx -39.284726889$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3+10a by a.

$$-390a-\left(3a+10a^{2}\right)=6$$

To find the opposite of 3a+10a^{2}, find the opposite of each term.

$$-390a-3a-10a^{2}=6$$

Combine -390a and -3a to get -393a.

$$-393a-10a^{2}=6$$

Subtract 6 from both sides.

$$-393a-10a^{2}-6=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-10a^{2}-393a-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, -393 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-393\right)±\sqrt{\left(-393\right)^{2}-4\left(-10\right)\left(-6\right)}}{2\left(-10\right)}$$

Square -393.

$$a=\frac{-\left(-393\right)±\sqrt{154449-4\left(-10\right)\left(-6\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$a=\frac{-\left(-393\right)±\sqrt{154449+40\left(-6\right)}}{2\left(-10\right)}$$

Multiply 40 times -6.

$$a=\frac{-\left(-393\right)±\sqrt{154449-240}}{2\left(-10\right)}$$

Add 154449 to -240.

$$a=\frac{-\left(-393\right)±\sqrt{154209}}{2\left(-10\right)}$$

The opposite of -393 is 393.

$$a=\frac{393±\sqrt{154209}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$a=\frac{393±\sqrt{154209}}{-20}$$

Now solve the equation a=\frac{393±\sqrt{154209}}{-20} when ± is plus. Add 393 to \sqrt{154209}.

$$a=\frac{\sqrt{154209}+393}{-20}$$

Divide 393+\sqrt{154209} by -20.

$$a=\frac{-\sqrt{154209}-393}{20}$$

Now solve the equation a=\frac{393±\sqrt{154209}}{-20} when ± is minus. Subtract \sqrt{154209} from 393.

$$a=\frac{393-\sqrt{154209}}{-20}$$

Divide 393-\sqrt{154209} by -20.

$$a=\frac{\sqrt{154209}-393}{20}$$

The equation is now solved.

$$a=\frac{-\sqrt{154209}-393}{20} a=\frac{\sqrt{154209}-393}{20}$$

Steps for Completing the Square

Use the distributive property to multiply 3+10a by a.

$$-390a-\left(3a+10a^{2}\right)=6$$

To find the opposite of 3a+10a^{2}, find the opposite of each term.

$$-390a-3a-10a^{2}=6$$

Combine -390a and -3a to get -393a.

$$-393a-10a^{2}=6$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-10a^{2}-393a=6$$

Divide both sides by -10.

$$\frac{-10a^{2}-393a}{-10}=\frac{6}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$a^{2}+\left(-\frac{393}{-10}\right)a=\frac{6}{-10}$$

Divide -393 by -10.

$$a^{2}+\frac{393}{10}a=\frac{6}{-10}$$

Reduce the fraction \frac{6}{-10} to lowest terms by extracting and canceling out 2.

$$a^{2}+\frac{393}{10}a=-\frac{3}{5}$$

Divide \frac{393}{10}, the coefficient of the x term, by 2 to get \frac{393}{20}. Then add the square of \frac{393}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{393}{10}a+\left(\frac{393}{20}\right)^{2}=-\frac{3}{5}+\left(\frac{393}{20}\right)^{2}$$

Square \frac{393}{20} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{393}{10}a+\frac{154449}{400}=-\frac{3}{5}+\frac{154449}{400}$$

Add -\frac{3}{5} to \frac{154449}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{393}{10}a+\frac{154449}{400}=\frac{154209}{400}$$

Factor a^{2}+\frac{393}{10}a+\frac{154449}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{393}{20}\right)^{2}=\frac{154209}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{393}{20}\right)^{2}}=\sqrt{\frac{154209}{400}}$$

Simplify.

$$a+\frac{393}{20}=\frac{\sqrt{154209}}{20} a+\frac{393}{20}=-\frac{\sqrt{154209}}{20}$$

Subtract \frac{393}{20} from both sides of the equation.

$$a=\frac{\sqrt{154209}-393}{20} a=\frac{-\sqrt{154209}-393}{20}$$