Solve 10x^2-8.96+1.22x-78840=0

$$10 x ^ { 2 } - 8.96 + 1.22 x - 78840 = 0$$

Solve for x

$$x = \frac{27 \sqrt{10816049} - 61}{1000} \approx 88.735957836$$

$$x=\frac{-27\sqrt{10816049}-61}{1000}\approx -88.857957836$$

Steps Using the Quadratic Formula

Subtract 78840 from -8.96 to get -78848.96.

$$10x^{2}-78848.96+1.22x=0$$

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.

$$10x^{2}+1.22x-78848.96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 1.22 for b, and -78848.96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.22±\sqrt{1.22^{2}-4\times 10\left(-78848.96\right)}}{2\times 10}$$

Square 1.22 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.22±\sqrt{1.4884-4\times 10\left(-78848.96\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-1.22±\sqrt{1.4884-40\left(-78848.96\right)}}{2\times 10}$$

Multiply -40 times -78848.96.

$$x=\frac{-1.22±\sqrt{1.4884+3153958.4}}{2\times 10}$$

Add 1.4884 to 3153958.4 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.22±\sqrt{3153959.8884}}{2\times 10}$$

Take the square root of 3153959.8884.

$$x=\frac{-1.22±\frac{27\sqrt{10816049}}{50}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-1.22±\frac{27\sqrt{10816049}}{50}}{20}$$

Now solve the equation x=\frac{-1.22±\frac{27\sqrt{10816049}}{50}}{20} when ± is plus. Add -1.22 to \frac{27\sqrt{10816049}}{50}.

$$x=\frac{27\sqrt{10816049}-61}{20\times 50}$$

Divide \frac{-61+27\sqrt{10816049}}{50} by 20.

$$x=\frac{27\sqrt{10816049}-61}{1000}$$

Now solve the equation x=\frac{-1.22±\frac{27\sqrt{10816049}}{50}}{20} when ± is minus. Subtract \frac{27\sqrt{10816049}}{50} from -1.22.

$$x=\frac{-27\sqrt{10816049}-61}{20\times 50}$$

Divide \frac{-61-27\sqrt{10816049}}{50} by 20.

$$x=\frac{-27\sqrt{10816049}-61}{1000}$$

The equation is now solved.

$$x=\frac{27\sqrt{10816049}-61}{1000} x=\frac{-27\sqrt{10816049}-61}{1000}$$

Steps for Completing the Square

Subtract 78840 from -8.96 to get -78848.96.

$$10x^{2}-78848.96+1.22x=0$$

Add 78848.96 to both sides. Anything plus zero gives itself.

$$10x^{2}+1.22x=78848.96$$

Divide both sides by 10.

$$\frac{10x^{2}+1.22x}{10}=\frac{78848.96}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{1.22}{10}x=\frac{78848.96}{10}$$

Divide 1.22 by 10.

$$x^{2}+0.122x=\frac{78848.96}{10}$$

Divide 78848.96 by 10.

$$x^{2}+0.122x=7884.896$$

Divide 0.122, the coefficient of the x term, by 2 to get 0.061. Then add the square of 0.061 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.122x+0.061^{2}=7884.896+0.061^{2}$$

Square 0.061 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.122x+0.003721=7884.896+0.003721$$

Add 7884.896 to 0.003721 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+0.122x+0.003721=7884.899721$$

Factor x^{2}+0.122x+0.003721. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.061\right)^{2}=7884.899721$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.061\right)^{2}}=\sqrt{7884.899721}$$

Simplify.

$$x+0.061=\frac{27\sqrt{10816049}}{1000} x+0.061=-\frac{27\sqrt{10816049}}{1000}$$

Subtract 0.061 from both sides of the equation.

$$x=\frac{27\sqrt{10816049}-61}{1000} x=\frac{-27\sqrt{10816049}-61}{1000}$$