Solve a-4=14/a

$$a - 4 = \frac { 14 } { a }$$

Solve for a

$$a=3\sqrt{2}+2\approx 6.242640687$$

$$a=2-3\sqrt{2}\approx -2.242640687$$

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.

$$aa+a\left(-4\right)=14$$

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

$$a^{2}+a\left(-4\right)=14$$

Subtract 14 from both sides.

$$a^{2}+a\left(-4\right)-14=0$$

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.

$$a^{2}-4a-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-14\right)}}{2}$$

Square -4.

$$a=\frac{-\left(-4\right)±\sqrt{16-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$a=\frac{-\left(-4\right)±\sqrt{16+56}}{2}$$

Add 16 to 56.

$$a=\frac{-\left(-4\right)±\sqrt{72}}{2}$$

Take the square root of 72.

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

The opposite of -4 is 4.

$$a=\frac{4±6\sqrt{2}}{2}$$

Now solve the equation a=\frac{4±6\sqrt{2}}{2} when ± is plus. Add 4 to 6\sqrt{2}.

$$a=\frac{6\sqrt{2}+4}{2}$$

Divide 4+6\sqrt{2} by 2.

$$a=3\sqrt{2}+2$$

Now solve the equation a=\frac{4±6\sqrt{2}}{2} when ± is minus. Subtract 6\sqrt{2} from 4.

$$a=\frac{4-6\sqrt{2}}{2}$$

Divide 4-6\sqrt{2} by 2.

$$a=2-3\sqrt{2}$$

The equation is now solved.

$$a=3\sqrt{2}+2 a=2-3\sqrt{2}$$

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.

$$aa+a\left(-4\right)=14$$

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

$$a^{2}+a\left(-4\right)=14$$

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.

$$a^{2}-4a=14$$

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

$$a^{2}-4a+\left(-2\right)^{2}=14+\left(-2\right)^{2}$$

Square -2.

$$a^{2}-4a+4=14+4$$

Add 14 to 4.

$$a^{2}-4a+4=18$$

Factor a^{2}-4a+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}.

$$\left(a-2\right)^{2}=18$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2\right)^{2}}=\sqrt{18}$$

Simplify.

$$a-2=3\sqrt{2} a-2=-3\sqrt{2}$$

Add 2 to both sides of the equation.

$$a=3\sqrt{2}+2 a=2-3\sqrt{2}$$