Solve a^2-56a+16=144a^2

$${ a }^{ 2 } -56a+16 = 144 { a }^{ 2 }$$

Solve for a

$$a=\frac{32\sqrt{3}-28}{143}\approx 0.191787593$$

$$a=\frac{-32\sqrt{3}-28}{143}\approx -0.583395985$$

Steps Using the Quadratic Formula

Subtract 144a^{2} from both sides.

$$a^{2}-56a+16-144a^{2}=0$$

Combine a^{2} and -144a^{2} to get -143a^{2}.

$$-143a^{2}-56a+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -143 for a, -56 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\left(-143\right)\times 16}}{2\left(-143\right)}$$

Square -56.

$$a=\frac{-\left(-56\right)±\sqrt{3136-4\left(-143\right)\times 16}}{2\left(-143\right)}$$

Multiply -4 times -143.

$$a=\frac{-\left(-56\right)±\sqrt{3136+572\times 16}}{2\left(-143\right)}$$

Multiply 572 times 16.

$$a=\frac{-\left(-56\right)±\sqrt{3136+9152}}{2\left(-143\right)}$$

Add 3136 to 9152.

$$a=\frac{-\left(-56\right)±\sqrt{12288}}{2\left(-143\right)}$$

Take the square root of 12288.

$$a=\frac{-\left(-56\right)±64\sqrt{3}}{2\left(-143\right)}$$

The opposite of -56 is 56.

$$a=\frac{56±64\sqrt{3}}{2\left(-143\right)}$$

Multiply 2 times -143.

$$a=\frac{56±64\sqrt{3}}{-286}$$

Now solve the equation a=\frac{56±64\sqrt{3}}{-286} when ± is plus. Add 56 to 64\sqrt{3}.

$$a=\frac{64\sqrt{3}+56}{-286}$$

Divide 56+64\sqrt{3} by -286.

$$a=\frac{-32\sqrt{3}-28}{143}$$

Now solve the equation a=\frac{56±64\sqrt{3}}{-286} when ± is minus. Subtract 64\sqrt{3} from 56.

$$a=\frac{56-64\sqrt{3}}{-286}$$

Divide 56-64\sqrt{3} by -286.

$$a=\frac{32\sqrt{3}-28}{143}$$

The equation is now solved.

$$a=\frac{-32\sqrt{3}-28}{143} a=\frac{32\sqrt{3}-28}{143}$$

Steps for Completing the Square

Subtract 144a^{2} from both sides.

$$a^{2}-56a+16-144a^{2}=0$$

Combine a^{2} and -144a^{2} to get -143a^{2}.

$$-143a^{2}-56a+16=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$-143a^{2}-56a=-16$$

Divide both sides by -143.

$$\frac{-143a^{2}-56a}{-143}=-\frac{16}{-143}$$

Dividing by -143 undoes the multiplication by -143.

$$a^{2}+\left(-\frac{56}{-143}\right)a=-\frac{16}{-143}$$

Divide -56 by -143.

$$a^{2}+\frac{56}{143}a=-\frac{16}{-143}$$

Divide -16 by -143.

$$a^{2}+\frac{56}{143}a=\frac{16}{143}$$

Divide \frac{56}{143}, the coefficient of the x term, by 2 to get \frac{28}{143}. Then add the square of \frac{28}{143} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{56}{143}a+\left(\frac{28}{143}\right)^{2}=\frac{16}{143}+\left(\frac{28}{143}\right)^{2}$$

Square \frac{28}{143} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{56}{143}a+\frac{784}{20449}=\frac{16}{143}+\frac{784}{20449}$$

Add \frac{16}{143} to \frac{784}{20449} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{56}{143}a+\frac{784}{20449}=\frac{3072}{20449}$$

Factor a^{2}+\frac{56}{143}a+\frac{784}{20449}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{28}{143}\right)^{2}=\frac{3072}{20449}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{28}{143}\right)^{2}}=\sqrt{\frac{3072}{20449}}$$

Simplify.

$$a+\frac{28}{143}=\frac{32\sqrt{3}}{143} a+\frac{28}{143}=-\frac{32\sqrt{3}}{143}$$

Subtract \frac{28}{143} from both sides of the equation.

$$a=\frac{32\sqrt{3}-28}{143} a=\frac{-32\sqrt{3}-28}{143}$$