Solve 92a^2+562a+75=60

$$92 a ^ { 2 } + 562 a + 75 = 60$$

Solve for a

$$a=\frac{\sqrt{77581}-281}{92}\approx -0.026808039$$

$$a=\frac{-\sqrt{77581}-281}{92}\approx -6.081887614$$

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.

$$92a^{2}+562a+75=60$$

Subtract 60 from both sides of the equation.

$$92a^{2}+562a+75-60=60-60$$

Subtracting 60 from itself leaves 0.

$$92a^{2}+562a+75-60=0$$

Subtract 60 from 75.

$$92a^{2}+562a+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 92 for a, 562 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-562±\sqrt{562^{2}-4\times 92\times 15}}{2\times 92}$$

Square 562.

$$a=\frac{-562±\sqrt{315844-4\times 92\times 15}}{2\times 92}$$

Multiply -4 times 92.

$$a=\frac{-562±\sqrt{315844-368\times 15}}{2\times 92}$$

Multiply -368 times 15.

$$a=\frac{-562±\sqrt{315844-5520}}{2\times 92}$$

Add 315844 to -5520.

$$a=\frac{-562±\sqrt{310324}}{2\times 92}$$

Take the square root of 310324.

$$a=\frac{-562±2\sqrt{77581}}{2\times 92}$$

Multiply 2 times 92.

$$a=\frac{-562±2\sqrt{77581}}{184}$$

Now solve the equation a=\frac{-562±2\sqrt{77581}}{184} when ± is plus. Add -562 to 2\sqrt{77581}.

$$a=\frac{2\sqrt{77581}-562}{184}$$

Divide -562+2\sqrt{77581} by 184.

$$a=\frac{\sqrt{77581}-281}{92}$$

Now solve the equation a=\frac{-562±2\sqrt{77581}}{184} when ± is minus. Subtract 2\sqrt{77581} from -562.

$$a=\frac{-2\sqrt{77581}-562}{184}$$

Divide -562-2\sqrt{77581} by 184.

$$a=\frac{-\sqrt{77581}-281}{92}$$

The equation is now solved.

$$a=\frac{\sqrt{77581}-281}{92} a=\frac{-\sqrt{77581}-281}{92}$$

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.

$$92a^{2}+562a+75=60$$

Subtract 75 from both sides of the equation.

$$92a^{2}+562a+75-75=60-75$$

Subtracting 75 from itself leaves 0.

$$92a^{2}+562a=60-75$$

Subtract 75 from 60.

$$92a^{2}+562a=-15$$

Divide both sides by 92.

$$\frac{92a^{2}+562a}{92}=-\frac{15}{92}$$

Dividing by 92 undoes the multiplication by 92.

$$a^{2}+\frac{562}{92}a=-\frac{15}{92}$$

Reduce the fraction \frac{562}{92} to lowest terms by extracting and canceling out 2.

$$a^{2}+\frac{281}{46}a=-\frac{15}{92}$$

Divide \frac{281}{46}, the coefficient of the x term, by 2 to get \frac{281}{92}. Then add the square of \frac{281}{92} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{281}{46}a+\left(\frac{281}{92}\right)^{2}=-\frac{15}{92}+\left(\frac{281}{92}\right)^{2}$$

Square \frac{281}{92} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{281}{46}a+\frac{78961}{8464}=-\frac{15}{92}+\frac{78961}{8464}$$

Add -\frac{15}{92} to \frac{78961}{8464} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{281}{46}a+\frac{78961}{8464}=\frac{77581}{8464}$$

Factor a^{2}+\frac{281}{46}a+\frac{78961}{8464}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{281}{92}\right)^{2}=\frac{77581}{8464}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{281}{92}\right)^{2}}=\sqrt{\frac{77581}{8464}}$$

Simplify.

$$a+\frac{281}{92}=\frac{\sqrt{77581}}{92} a+\frac{281}{92}=-\frac{\sqrt{77581}}{92}$$

Subtract \frac{281}{92} from both sides of the equation.

$$a=\frac{\sqrt{77581}-281}{92} a=\frac{-\sqrt{77581}-281}{92}$$