Solve 3a+14/2a-2-2a=7

$$\frac { 3 a + 14 } { 2 a - 2 } - 2 a = 7$$

Solve for a

$$a = \frac{\sqrt{497} - 7}{8} \approx 1.911687101$$

$$a=\frac{-\sqrt{497}-7}{8}\approx -3.661687101$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 2\left(a-1\right).

$$3a+14-2a\times 2\left(a-1\right)=14\left(a-1\right)$$

Multiply -2 and 2 to get -4.

$$3a+14-4a\left(a-1\right)=14\left(a-1\right)$$

Use the distributive property to multiply -4a by a-1.

$$3a+14-4a^{2}+4a=14\left(a-1\right)$$

Combine 3a and 4a to get 7a.

$$7a+14-4a^{2}=14\left(a-1\right)$$

Use the distributive property to multiply 14 by a-1.

$$7a+14-4a^{2}=14a-14$$

Subtract 14a from both sides.

$$7a+14-4a^{2}-14a=-14$$

Combine 7a and -14a to get -7a.

$$-7a+14-4a^{2}=-14$$

Add 14 to both sides.

$$-7a+14-4a^{2}+14=0$$

Add 14 and 14 to get 28.

$$-7a+28-4a^{2}=0$$

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.

$$-4a^{2}-7a+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -7 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-4\right)\times 28}}{2\left(-4\right)}$$

Square -7.

$$a=\frac{-\left(-7\right)±\sqrt{49-4\left(-4\right)\times 28}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$a=\frac{-\left(-7\right)±\sqrt{49+16\times 28}}{2\left(-4\right)}$$

Multiply 16 times 28.

$$a=\frac{-\left(-7\right)±\sqrt{49+448}}{2\left(-4\right)}$$

Add 49 to 448.

$$a=\frac{-\left(-7\right)±\sqrt{497}}{2\left(-4\right)}$$

The opposite of -7 is 7.

$$a=\frac{7±\sqrt{497}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$a=\frac{7±\sqrt{497}}{-8}$$

Now solve the equation a=\frac{7±\sqrt{497}}{-8} when ± is plus. Add 7 to \sqrt{497}.

$$a=\frac{\sqrt{497}+7}{-8}$$

Divide 7+\sqrt{497} by -8.

$$a=\frac{-\sqrt{497}-7}{8}$$

Now solve the equation a=\frac{7±\sqrt{497}}{-8} when ± is minus. Subtract \sqrt{497} from 7.

$$a=\frac{7-\sqrt{497}}{-8}$$

Divide 7-\sqrt{497} by -8.

$$a=\frac{\sqrt{497}-7}{8}$$

The equation is now solved.

$$a=\frac{-\sqrt{497}-7}{8} a=\frac{\sqrt{497}-7}{8}$$

Steps for Completing the Square

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 2\left(a-1\right).

$$3a+14-2a\times 2\left(a-1\right)=14\left(a-1\right)$$

Multiply -2 and 2 to get -4.

$$3a+14-4a\left(a-1\right)=14\left(a-1\right)$$

Use the distributive property to multiply -4a by a-1.

$$3a+14-4a^{2}+4a=14\left(a-1\right)$$

Combine 3a and 4a to get 7a.

$$7a+14-4a^{2}=14\left(a-1\right)$$

Use the distributive property to multiply 14 by a-1.

$$7a+14-4a^{2}=14a-14$$

Subtract 14a from both sides.

$$7a+14-4a^{2}-14a=-14$$

Combine 7a and -14a to get -7a.

$$-7a+14-4a^{2}=-14$$

Subtract 14 from both sides.

$$-7a-4a^{2}=-14-14$$

Subtract 14 from -14 to get -28.

$$-7a-4a^{2}=-28$$

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.

$$-4a^{2}-7a=-28$$

Divide both sides by -4.

$$\frac{-4a^{2}-7a}{-4}=-\frac{28}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$a^{2}+\left(-\frac{7}{-4}\right)a=-\frac{28}{-4}$$

Divide -7 by -4.

$$a^{2}+\frac{7}{4}a=-\frac{28}{-4}$$

Divide -28 by -4.

$$a^{2}+\frac{7}{4}a=7$$

Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{7}{4}a+\left(\frac{7}{8}\right)^{2}=7+\left(\frac{7}{8}\right)^{2}$$

Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{7}{4}a+\frac{49}{64}=7+\frac{49}{64}$$

Add 7 to \frac{49}{64}.

$$a^{2}+\frac{7}{4}a+\frac{49}{64}=\frac{497}{64}$$

Factor a^{2}+\frac{7}{4}a+\frac{49}{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}.

$$\left(a+\frac{7}{8}\right)^{2}=\frac{497}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{7}{8}\right)^{2}}=\sqrt{\frac{497}{64}}$$

Simplify.

$$a+\frac{7}{8}=\frac{\sqrt{497}}{8} a+\frac{7}{8}=-\frac{\sqrt{497}}{8}$$

Subtract \frac{7}{8} from both sides of the equation.

$$a=\frac{\sqrt{497}-7}{8} a=\frac{-\sqrt{497}-7}{8}$$