Solve 9a^2-78a+170=0

$$9 { a }^{ 2 } -78a+170 = 0$$

Solve for a

$$a=\frac{13}{3}+\frac{1}{3}i\approx 4.333333333+0.333333333i$$

$$a=\frac{13}{3}-\frac{1}{3}i\approx 4.333333333-0.333333333i$$

Steps Using the Quadratic Formula

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.

$$9a^{2}-78a+170=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -78 for b, and 170 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-78\right)±\sqrt{\left(-78\right)^{2}-4\times 9\times 170}}{2\times 9}$$

Square -78.

$$a=\frac{-\left(-78\right)±\sqrt{6084-4\times 9\times 170}}{2\times 9}$$

Multiply -4 times 9.

$$a=\frac{-\left(-78\right)±\sqrt{6084-36\times 170}}{2\times 9}$$

Multiply -36 times 170.

$$a=\frac{-\left(-78\right)±\sqrt{6084-6120}}{2\times 9}$$

Add 6084 to -6120.

$$a=\frac{-\left(-78\right)±\sqrt{-36}}{2\times 9}$$

Take the square root of -36.

$$a=\frac{-\left(-78\right)±6i}{2\times 9}$$

The opposite of -78 is 78.

$$a=\frac{78±6i}{2\times 9}$$

Multiply 2 times 9.

$$a=\frac{78±6i}{18}$$

Now solve the equation a=\frac{78±6i}{18} when ± is plus. Add 78 to 6i.

$$a=\frac{78+6i}{18}$$

Divide 78+6i by 18.

$$a=\frac{13}{3}+\frac{1}{3}i$$

Now solve the equation a=\frac{78±6i}{18} when ± is minus. Subtract 6i from 78.

$$a=\frac{78-6i}{18}$$

Divide 78-6i by 18.

$$a=\frac{13}{3}-\frac{1}{3}i$$

The equation is now solved.

$$a=\frac{13}{3}+\frac{1}{3}i a=\frac{13}{3}-\frac{1}{3}i$$

Steps for Completing the Square

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.

$$9a^{2}-78a+170=0$$

Subtract 170 from both sides of the equation.

$$9a^{2}-78a+170-170=-170$$

Subtracting 170 from itself leaves 0.

$$9a^{2}-78a=-170$$

Divide both sides by 9.

$$\frac{9a^{2}-78a}{9}=-\frac{170}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$a^{2}+\left(-\frac{78}{9}\right)a=-\frac{170}{9}$$

Reduce the fraction \frac{-78}{9} to lowest terms by extracting and canceling out 3.

$$a^{2}-\frac{26}{3}a=-\frac{170}{9}$$

Divide -\frac{26}{3}, the coefficient of the x term, by 2 to get -\frac{13}{3}. Then add the square of -\frac{13}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{26}{3}a+\left(-\frac{13}{3}\right)^{2}=-\frac{170}{9}+\left(-\frac{13}{3}\right)^{2}$$

Square -\frac{13}{3} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{26}{3}a+\frac{169}{9}=\frac{-170+169}{9}$$

Add -\frac{170}{9} to \frac{169}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{26}{3}a+\frac{169}{9}=-\frac{1}{9}$$

Factor a^{2}-\frac{26}{3}a+\frac{169}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{13}{3}\right)^{2}=-\frac{1}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{13}{3}\right)^{2}}=\sqrt{-\frac{1}{9}}$$

Simplify.

$$a-\frac{13}{3}=\frac{1}{3}i a-\frac{13}{3}=-\frac{1}{3}i$$

Add \frac{13}{3} to both sides of the equation.

$$a=\frac{13}{3}+\frac{1}{3}i a=\frac{13}{3}-\frac{1}{3}i$$