Solve 50a^2+96=10a^2-136a

$$50 a ^ { 2 } + 96 = 10 a ^ { 2 } - 136 a$$

Solve for a

$$a = -\frac{12}{5} = -2\frac{2}{5} = -2.4$$

$$a=-1$$

Steps Using the Quadratic Formula

Subtract 10a^{2} from both sides.

$$50a^{2}+96-10a^{2}=-136a$$

Combine 50a^{2} and -10a^{2} to get 40a^{2}.

$$40a^{2}+96=-136a$$

Add 136a to both sides.

$$40a^{2}+96+136a=0$$

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.

$$40a^{2}+136a+96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 40 for a, 136 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-136±\sqrt{136^{2}-4\times 40\times 96}}{2\times 40}$$

Square 136.

$$a=\frac{-136±\sqrt{18496-4\times 40\times 96}}{2\times 40}$$

Multiply -4 times 40.

$$a=\frac{-136±\sqrt{18496-160\times 96}}{2\times 40}$$

Multiply -160 times 96.

$$a=\frac{-136±\sqrt{18496-15360}}{2\times 40}$$

Add 18496 to -15360.

$$a=\frac{-136±\sqrt{3136}}{2\times 40}$$

Take the square root of 3136.

$$a=\frac{-136±56}{2\times 40}$$

Multiply 2 times 40.

$$a=\frac{-136±56}{80}$$

Now solve the equation a=\frac{-136±56}{80} when ± is plus. Add -136 to 56.

$$a=-\frac{80}{80}$$

Divide -80 by 80.

$$a=-1$$

Now solve the equation a=\frac{-136±56}{80} when ± is minus. Subtract 56 from -136.

$$a=-\frac{192}{80}$$

Reduce the fraction \frac{-192}{80} to lowest terms by extracting and canceling out 16.

$$a=-\frac{12}{5}$$

The equation is now solved.

$$a=-1 a=-\frac{12}{5}$$

Steps for Completing the Square

Subtract 10a^{2} from both sides.

$$50a^{2}+96-10a^{2}=-136a$$

Combine 50a^{2} and -10a^{2} to get 40a^{2}.

$$40a^{2}+96=-136a$$

Add 136a to both sides.

$$40a^{2}+96+136a=0$$

Subtract 96 from both sides. Anything subtracted from zero gives its negation.

$$40a^{2}+136a=-96$$

Divide both sides by 40.

$$\frac{40a^{2}+136a}{40}=-\frac{96}{40}$$

Dividing by 40 undoes the multiplication by 40.

$$a^{2}+\frac{136}{40}a=-\frac{96}{40}$$

Reduce the fraction \frac{136}{40} to lowest terms by extracting and canceling out 8.

$$a^{2}+\frac{17}{5}a=-\frac{96}{40}$$

Reduce the fraction \frac{-96}{40} to lowest terms by extracting and canceling out 8.

$$a^{2}+\frac{17}{5}a=-\frac{12}{5}$$

Divide \frac{17}{5}, the coefficient of the x term, by 2 to get \frac{17}{10}. Then add the square of \frac{17}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{17}{5}a+\left(\frac{17}{10}\right)^{2}=-\frac{12}{5}+\left(\frac{17}{10}\right)^{2}$$

Square \frac{17}{10} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{17}{5}a+\frac{289}{100}=-\frac{12}{5}+\frac{289}{100}$$

Add -\frac{12}{5} to \frac{289}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{17}{5}a+\frac{289}{100}=\frac{49}{100}$$

Factor a^{2}+\frac{17}{5}a+\frac{289}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{17}{10}\right)^{2}=\frac{49}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{17}{10}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Simplify.

$$a+\frac{17}{10}=\frac{7}{10} a+\frac{17}{10}=-\frac{7}{10}$$

Subtract \frac{17}{10} from both sides of the equation.

$$a=-1 a=-\frac{12}{5}$$