Solve 5a^2-a+1=3a+5

$$5 a ^ { 2 } - a + 1 = 3 a + 5$$

Solve for a

$$a = \frac{2 \sqrt{6} + 2}{5} \approx 1.379795897$$

$$a=\frac{2-2\sqrt{6}}{5}\approx -0.579795897$$

Steps Using the Quadratic Formula

Subtract 3a from both sides.

$$5a^{2}-a+1-3a=5$$

Combine -a and -3a to get -4a.

$$5a^{2}-4a+1=5$$

Subtract 5 from both sides.

$$5a^{2}-4a+1-5=0$$

Subtract 5 from 1 to get -4.

$$5a^{2}-4a-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -4 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 5\left(-4\right)}}{2\times 5}$$

Square -4.

$$a=\frac{-\left(-4\right)±\sqrt{16-4\times 5\left(-4\right)}}{2\times 5}$$

Multiply -4 times 5.

$$a=\frac{-\left(-4\right)±\sqrt{16-20\left(-4\right)}}{2\times 5}$$

Multiply -20 times -4.

$$a=\frac{-\left(-4\right)±\sqrt{16+80}}{2\times 5}$$

Add 16 to 80.

$$a=\frac{-\left(-4\right)±\sqrt{96}}{2\times 5}$$

Take the square root of 96.

$$a=\frac{-\left(-4\right)±4\sqrt{6}}{2\times 5}$$

The opposite of -4 is 4.

$$a=\frac{4±4\sqrt{6}}{2\times 5}$$

Multiply 2 times 5.

$$a=\frac{4±4\sqrt{6}}{10}$$

Now solve the equation a=\frac{4±4\sqrt{6}}{10} when ± is plus. Add 4 to 4\sqrt{6}.

$$a=\frac{4\sqrt{6}+4}{10}$$

Divide 4+4\sqrt{6} by 10.

$$a=\frac{2\sqrt{6}+2}{5}$$

Now solve the equation a=\frac{4±4\sqrt{6}}{10} when ± is minus. Subtract 4\sqrt{6} from 4.

$$a=\frac{4-4\sqrt{6}}{10}$$

Divide 4-4\sqrt{6} by 10.

$$a=\frac{2-2\sqrt{6}}{5}$$

The equation is now solved.

$$a=\frac{2\sqrt{6}+2}{5} a=\frac{2-2\sqrt{6}}{5}$$

Steps for Completing the Square

Subtract 3a from both sides.

$$5a^{2}-a+1-3a=5$$

Combine -a and -3a to get -4a.

$$5a^{2}-4a+1=5$$

Subtract 1 from both sides.

$$5a^{2}-4a=5-1$$

Subtract 1 from 5 to get 4.

$$5a^{2}-4a=4$$

Divide both sides by 5.

$$\frac{5a^{2}-4a}{5}=\frac{4}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$a^{2}-\frac{4}{5}a=\frac{4}{5}$$

Divide -\frac{4}{5}, the coefficient of the x term, by 2 to get -\frac{2}{5}. Then add the square of -\frac{2}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{4}{5}a+\left(-\frac{2}{5}\right)^{2}=\frac{4}{5}+\left(-\frac{2}{5}\right)^{2}$$

Square -\frac{2}{5} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{4}{5}a+\frac{4}{25}=\frac{4}{5}+\frac{4}{25}$$

Add \frac{4}{5} to \frac{4}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{4}{5}a+\frac{4}{25}=\frac{24}{25}$$

Factor a^{2}-\frac{4}{5}a+\frac{4}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{2}{5}\right)^{2}=\frac{24}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{2}{5}\right)^{2}}=\sqrt{\frac{24}{25}}$$

Simplify.

$$a-\frac{2}{5}=\frac{2\sqrt{6}}{5} a-\frac{2}{5}=-\frac{2\sqrt{6}}{5}$$

Add \frac{2}{5} to both sides of the equation.

$$a=\frac{2\sqrt{6}+2}{5} a=\frac{2-2\sqrt{6}}{5}$$