Solve 5a^2-a-5a+1=12a^2-5a-6a

$$5 a ^ { 2 } - a - 5 a + 1 = 12 a ^ { 2 } - 5 a - 6 a$$

Solve for a

$$a=\frac{\sqrt{53}+5}{14}\approx 0.877150706$$

$$a=\frac{5-\sqrt{53}}{14}\approx -0.162864992$$

Steps Using the Quadratic Formula

Combine -a and -5a to get -6a.

$$5a^{2}-6a+1=12a^{2}-5a-6a$$

Combine -5a and -6a to get -11a.

$$5a^{2}-6a+1=12a^{2}-11a$$

Subtract 12a^{2} from both sides.

$$5a^{2}-6a+1-12a^{2}=-11a$$

Combine 5a^{2} and -12a^{2} to get -7a^{2}.

$$-7a^{2}-6a+1=-11a$$

Add 11a to both sides.

$$-7a^{2}-6a+1+11a=0$$

Combine -6a and 11a to get 5a.

$$-7a^{2}+5a+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 5 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-5±\sqrt{5^{2}-4\left(-7\right)}}{2\left(-7\right)}$$

Square 5.

$$a=\frac{-5±\sqrt{25-4\left(-7\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$a=\frac{-5±\sqrt{25+28}}{2\left(-7\right)}$$

Add 25 to 28.

$$a=\frac{-5±\sqrt{53}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$a=\frac{-5±\sqrt{53}}{-14}$$

Now solve the equation a=\frac{-5±\sqrt{53}}{-14} when ± is plus. Add -5 to \sqrt{53}.

$$a=\frac{\sqrt{53}-5}{-14}$$

Divide -5+\sqrt{53} by -14.

$$a=\frac{5-\sqrt{53}}{14}$$

Now solve the equation a=\frac{-5±\sqrt{53}}{-14} when ± is minus. Subtract \sqrt{53} from -5.

$$a=\frac{-\sqrt{53}-5}{-14}$$

Divide -5-\sqrt{53} by -14.

$$a=\frac{\sqrt{53}+5}{14}$$

The equation is now solved.

$$a=\frac{5-\sqrt{53}}{14} a=\frac{\sqrt{53}+5}{14}$$

Steps for Completing the Square

Combine -a and -5a to get -6a.

$$5a^{2}-6a+1=12a^{2}-5a-6a$$

Combine -5a and -6a to get -11a.

$$5a^{2}-6a+1=12a^{2}-11a$$

Subtract 12a^{2} from both sides.

$$5a^{2}-6a+1-12a^{2}=-11a$$

Combine 5a^{2} and -12a^{2} to get -7a^{2}.

$$-7a^{2}-6a+1=-11a$$

Add 11a to both sides.

$$-7a^{2}-6a+1+11a=0$$

Combine -6a and 11a to get 5a.

$$-7a^{2}+5a+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-7a^{2}+5a=-1$$

Divide both sides by -7.

$$\frac{-7a^{2}+5a}{-7}=-\frac{1}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$a^{2}+\frac{5}{-7}a=-\frac{1}{-7}$$

Divide 5 by -7.

$$a^{2}-\frac{5}{7}a=-\frac{1}{-7}$$

Divide -1 by -7.

$$a^{2}-\frac{5}{7}a=\frac{1}{7}$$

Divide -\frac{5}{7}, the coefficient of the x term, by 2 to get -\frac{5}{14}. Then add the square of -\frac{5}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{5}{7}a+\left(-\frac{5}{14}\right)^{2}=\frac{1}{7}+\left(-\frac{5}{14}\right)^{2}$$

Square -\frac{5}{14} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{5}{7}a+\frac{25}{196}=\frac{1}{7}+\frac{25}{196}$$

Add \frac{1}{7} to \frac{25}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{5}{7}a+\frac{25}{196}=\frac{53}{196}$$

Factor a^{2}-\frac{5}{7}a+\frac{25}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{5}{14}\right)^{2}=\frac{53}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{5}{14}\right)^{2}}=\sqrt{\frac{53}{196}}$$

Simplify.

$$a-\frac{5}{14}=\frac{\sqrt{53}}{14} a-\frac{5}{14}=-\frac{\sqrt{53}}{14}$$

Add \frac{5}{14} to both sides of the equation.

$$a=\frac{\sqrt{53}+5}{14} a=\frac{5-\sqrt{53}}{14}$$