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-176x-121=64x^{2}
Subtract 121 from both sides.
-176x-121-64x^{2}=0
Subtract 64x^{2} from both sides.
-64x^{2}-176x-121=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.
x=\frac{-\left(-176\right)±\sqrt{\left(-176\right)^{2}-4\left(-64\right)\left(-121\right)}}{2\left(-64\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -64 for a, -176 for b, and -121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-176\right)±\sqrt{30976-4\left(-64\right)\left(-121\right)}}{2\left(-64\right)}
Square -176.
x=\frac{-\left(-176\right)±\sqrt{30976+256\left(-121\right)}}{2\left(-64\right)}
Multiply -4 times -64.
x=\frac{-\left(-176\right)±\sqrt{30976-30976}}{2\left(-64\right)}
Multiply 256 times -121.
x=\frac{-\left(-176\right)±\sqrt{0}}{2\left(-64\right)}
Add 30976 to -30976.
x=-\frac{-176}{2\left(-64\right)}
Take the square root of 0.
x=\frac{176}{2\left(-64\right)}
The opposite of -176 is 176.
x=\frac{176}{-128}
Multiply 2 times -64.
x=-\frac{11}{8}
Reduce the fraction \frac{176}{-128} to lowest terms by extracting and canceling out 16.
-176x-64x^{2}=121
Subtract 64x^{2} from both sides.
-64x^{2}-176x=121
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.
\frac{-64x^{2}-176x}{-64}=\frac{121}{-64}
Divide both sides by -64.
x^{2}+\left(-\frac{176}{-64}\right)x=\frac{121}{-64}
Dividing by -64 undoes the multiplication by -64.
x^{2}+\frac{11}{4}x=\frac{121}{-64}
Reduce the fraction \frac{-176}{-64} to lowest terms by extracting and canceling out 16.
x^{2}+\frac{11}{4}x=-\frac{121}{64}
Divide 121 by -64.
x^{2}+\frac{11}{4}x+\left(\frac{11}{8}\right)^{2}=-\frac{121}{64}+\left(\frac{11}{8}\right)^{2}
Divide \frac{11}{4}, the coefficient of the x term, by 2 to get \frac{11}{8}. Then add the square of \frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{11}{4}x+\frac{121}{64}=\frac{-121+121}{64}
Square \frac{11}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{11}{4}x+\frac{121}{64}=0
Add -\frac{121}{64} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{11}{8}\right)^{2}=0
Factor x^{2}+\frac{11}{4}x+\frac{121}{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}.
\sqrt{\left(x+\frac{11}{8}\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x+\frac{11}{8}=0 x+\frac{11}{8}=0
Simplify.
x=-\frac{11}{8} x=-\frac{11}{8}
Subtract \frac{11}{8} from both sides of the equation.
x=-\frac{11}{8}
The equation is now solved. Solutions are the same.