Solve 4a-9=1/a

$$4 a - 9 = \frac { 1 } { a }$$

Solve for a

$$a = \frac{\sqrt{97} + 9}{8} \approx 2.356107225$$

$$a=\frac{9-\sqrt{97}}{8}\approx -0.106107225$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$4aa+a\left(-9\right)=1$$

Multiply a and a to get a^{2}.

$$4a^{2}+a\left(-9\right)=1$$

Subtract 1 from both sides.

$$4a^{2}+a\left(-9\right)-1=0$$

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.

$$4a^{2}-9a-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -9 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\left(-1\right)}}{2\times 4}$$

Square -9.

$$a=\frac{-\left(-9\right)±\sqrt{81-4\times 4\left(-1\right)}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-\left(-9\right)±\sqrt{81-16\left(-1\right)}}{2\times 4}$$

Multiply -16 times -1.

$$a=\frac{-\left(-9\right)±\sqrt{81+16}}{2\times 4}$$

Add 81 to 16.

$$a=\frac{-\left(-9\right)±\sqrt{97}}{2\times 4}$$

The opposite of -9 is 9.

$$a=\frac{9±\sqrt{97}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{9±\sqrt{97}}{8}$$

Now solve the equation a=\frac{9±\sqrt{97}}{8} when ± is plus. Add 9 to \sqrt{97}.

$$a=\frac{\sqrt{97}+9}{8}$$

Now solve the equation a=\frac{9±\sqrt{97}}{8} when ± is minus. Subtract \sqrt{97} from 9.

$$a=\frac{9-\sqrt{97}}{8}$$

The equation is now solved.

$$a=\frac{\sqrt{97}+9}{8} a=\frac{9-\sqrt{97}}{8}$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$4aa+a\left(-9\right)=1$$

Multiply a and a to get a^{2}.

$$4a^{2}+a\left(-9\right)=1$$

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.

$$4a^{2}-9a=1$$

Divide both sides by 4.

$$\frac{4a^{2}-9a}{4}=\frac{1}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$a^{2}-\frac{9}{4}a=\frac{1}{4}$$

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{9}{4}a+\left(-\frac{9}{8}\right)^{2}=\frac{1}{4}+\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{9}{4}a+\frac{81}{64}=\frac{1}{4}+\frac{81}{64}$$

Add \frac{1}{4} to \frac{81}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{9}{4}a+\frac{81}{64}=\frac{97}{64}$$

Factor a^{2}-\frac{9}{4}a+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{9}{8}\right)^{2}=\frac{97}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{9}{8}\right)^{2}}=\sqrt{\frac{97}{64}}$$

Simplify.

$$a-\frac{9}{8}=\frac{\sqrt{97}}{8} a-\frac{9}{8}=-\frac{\sqrt{97}}{8}$$

Add \frac{9}{8} to both sides of the equation.

$$a=\frac{\sqrt{97}+9}{8} a=\frac{9-\sqrt{97}}{8}$$