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5x^{2}+12\left(1.44-0.96x+0.16x^{2}\right)-16x\left(-1.2+0.4x\right)=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-1.2+0.4x\right)^{2}.
5x^{2}+17.28-11.52x+1.92x^{2}-16x\left(-1.2+0.4x\right)=0
Use the distributive property to multiply 12 by 1.44-0.96x+0.16x^{2}.
6.92x^{2}+17.28-11.52x-16x\left(-1.2+0.4x\right)=0
Combine 5x^{2} and 1.92x^{2} to get 6.92x^{2}.
6.92x^{2}+17.28-11.52x+19.2x-6.4x^{2}=0
Use the distributive property to multiply -16x by -1.2+0.4x.
6.92x^{2}+17.28+7.68x-6.4x^{2}=0
Combine -11.52x and 19.2x to get 7.68x.
0.52x^{2}+17.28+7.68x=0
Combine 6.92x^{2} and -6.4x^{2} to get 0.52x^{2}.
0.52x^{2}+7.68x+17.28=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{-7.68±\sqrt{7.68^{2}-4\times 0.52\times 17.28}}{2\times 0.52}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.52 for a, 7.68 for b, and 17.28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7.68±\sqrt{58.9824-4\times 0.52\times 17.28}}{2\times 0.52}
Square 7.68 by squaring both the numerator and the denominator of the fraction.
x=\frac{-7.68±\sqrt{58.9824-2.08\times 17.28}}{2\times 0.52}
Multiply -4 times 0.52.
x=\frac{-7.68±\sqrt{\frac{36864-22464}{625}}}{2\times 0.52}
Multiply -2.08 times 17.28 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{-7.68±\sqrt{23.04}}{2\times 0.52}
Add 58.9824 to -35.9424 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{-7.68±\frac{24}{5}}{2\times 0.52}
Take the square root of 23.04.
x=\frac{-7.68±\frac{24}{5}}{1.04}
Multiply 2 times 0.52.
x=-\frac{\frac{72}{25}}{1.04}
Now solve the equation x=\frac{-7.68±\frac{24}{5}}{1.04} when ± is plus. Add -7.68 to \frac{24}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=-\frac{36}{13}
Divide -\frac{72}{25} by 1.04 by multiplying -\frac{72}{25} by the reciprocal of 1.04.
x=-\frac{\frac{312}{25}}{1.04}
Now solve the equation x=\frac{-7.68±\frac{24}{5}}{1.04} when ± is minus. Subtract \frac{24}{5} from -7.68 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
x=-12
Divide -\frac{312}{25} by 1.04 by multiplying -\frac{312}{25} by the reciprocal of 1.04.
x=-\frac{36}{13} x=-12
The equation is now solved.
5x^{2}+12\left(1.44-0.96x+0.16x^{2}\right)-16x\left(-1.2+0.4x\right)=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-1.2+0.4x\right)^{2}.
5x^{2}+17.28-11.52x+1.92x^{2}-16x\left(-1.2+0.4x\right)=0
Use the distributive property to multiply 12 by 1.44-0.96x+0.16x^{2}.
6.92x^{2}+17.28-11.52x-16x\left(-1.2+0.4x\right)=0
Combine 5x^{2} and 1.92x^{2} to get 6.92x^{2}.
6.92x^{2}+17.28-11.52x+19.2x-6.4x^{2}=0
Use the distributive property to multiply -16x by -1.2+0.4x.
6.92x^{2}+17.28+7.68x-6.4x^{2}=0
Combine -11.52x and 19.2x to get 7.68x.
0.52x^{2}+17.28+7.68x=0
Combine 6.92x^{2} and -6.4x^{2} to get 0.52x^{2}.
0.52x^{2}+7.68x=-17.28
Subtract 17.28 from both sides. Anything subtracted from zero gives its negation.
\frac{0.52x^{2}+7.68x}{0.52}=-\frac{17.28}{0.52}
Divide both sides of the equation by 0.52, which is the same as multiplying both sides by the reciprocal of the fraction.
x^{2}+\frac{7.68}{0.52}x=-\frac{17.28}{0.52}
Dividing by 0.52 undoes the multiplication by 0.52.
x^{2}+\frac{192}{13}x=-\frac{17.28}{0.52}
Divide 7.68 by 0.52 by multiplying 7.68 by the reciprocal of 0.52.
x^{2}+\frac{192}{13}x=-\frac{432}{13}
Divide -17.28 by 0.52 by multiplying -17.28 by the reciprocal of 0.52.
x^{2}+\frac{192}{13}x+\frac{96}{13}^{2}=-\frac{432}{13}+\frac{96}{13}^{2}
Divide \frac{192}{13}, the coefficient of the x term, by 2 to get \frac{96}{13}. Then add the square of \frac{96}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{192}{13}x+\frac{9216}{169}=-\frac{432}{13}+\frac{9216}{169}
Square \frac{96}{13} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{192}{13}x+\frac{9216}{169}=\frac{3600}{169}
Add -\frac{432}{13} to \frac{9216}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{96}{13}\right)^{2}=\frac{3600}{169}
Factor x^{2}+\frac{192}{13}x+\frac{9216}{169}. 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{96}{13}\right)^{2}}=\sqrt{\frac{3600}{169}}
Take the square root of both sides of the equation.
x+\frac{96}{13}=\frac{60}{13} x+\frac{96}{13}=-\frac{60}{13}
Simplify.
x=-\frac{36}{13} x=-12
Subtract \frac{96}{13} from both sides of the equation.