Solve 4183.92+2*78*9.8x=6.5*10^4quadx^2

$$4183.92 + 2 \times 78 \times 9.8 x = 6.5 \times 10 ^ { 4 } \quad x ^ { 2 }$$

Solve for x

$$x=\frac{3\sqrt{1119901}+147}{12500}\approx 0.2657409$$

$$x=\frac{147-3\sqrt{1119901}}{12500}\approx -0.2422209$$

Steps Using the Quadratic Formula

Multiply 2 and 78 to get 156.

$$4183.92+156\times 9.8x=6.5\times 10^{4}x^{2}$$

Multiply 156 and 9.8 to get 1528.8.

$$4183.92+1528.8x=6.5\times 10^{4}x^{2}$$

Calculate 10 to the power of 4 and get 10000.

$$4183.92+1528.8x=6.5\times 10000x^{2}$$

Multiply 6.5 and 10000 to get 65000.

$$4183.92+1528.8x=65000x^{2}$$

Subtract 65000x^{2} from both sides.

$$4183.92+1528.8x-65000x^{2}=0$$

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.

$$-65000x^{2}+1528.8x+4183.92=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -65000 for a, 1528.8 for b, and 4183.92 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1528.8±\sqrt{1528.8^{2}-4\left(-65000\right)\times 4183.92}}{2\left(-65000\right)}$$

Square 1528.8 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1528.8±\sqrt{2337229.44-4\left(-65000\right)\times 4183.92}}{2\left(-65000\right)}$$

Multiply -4 times -65000.

$$x=\frac{-1528.8±\sqrt{2337229.44+260000\times 4183.92}}{2\left(-65000\right)}$$

Multiply 260000 times 4183.92.

$$x=\frac{-1528.8±\sqrt{2337229.44+1087819200}}{2\left(-65000\right)}$$

Add 2337229.44 to 1087819200.

$$x=\frac{-1528.8±\sqrt{1090156429.44}}{2\left(-65000\right)}$$

Take the square root of 1090156429.44.

$$x=\frac{-1528.8±\frac{156\sqrt{1119901}}{5}}{2\left(-65000\right)}$$

Multiply 2 times -65000.

$$x=\frac{-1528.8±\frac{156\sqrt{1119901}}{5}}{-130000}$$

Now solve the equation x=\frac{-1528.8±\frac{156\sqrt{1119901}}{5}}{-130000} when ± is plus. Add -1528.8 to \frac{156\sqrt{1119901}}{5}.

$$x=\frac{156\sqrt{1119901}-7644}{-130000\times 5}$$

Divide \frac{-7644+156\sqrt{1119901}}{5} by -130000.

$$x=\frac{147-3\sqrt{1119901}}{12500}$$

Now solve the equation x=\frac{-1528.8±\frac{156\sqrt{1119901}}{5}}{-130000} when ± is minus. Subtract \frac{156\sqrt{1119901}}{5} from -1528.8.

$$x=\frac{-156\sqrt{1119901}-7644}{-130000\times 5}$$

Divide \frac{-7644-156\sqrt{1119901}}{5} by -130000.

$$x=\frac{3\sqrt{1119901}+147}{12500}$$

The equation is now solved.

$$x=\frac{147-3\sqrt{1119901}}{12500} x=\frac{3\sqrt{1119901}+147}{12500}$$

Steps for Completing the Square

Multiply 2 and 78 to get 156.

$$4183.92+156\times 9.8x=6.5\times 10^{4}x^{2}$$

Multiply 156 and 9.8 to get 1528.8.

$$4183.92+1528.8x=6.5\times 10^{4}x^{2}$$

Calculate 10 to the power of 4 and get 10000.

$$4183.92+1528.8x=6.5\times 10000x^{2}$$

Multiply 6.5 and 10000 to get 65000.

$$4183.92+1528.8x=65000x^{2}$$

Subtract 65000x^{2} from both sides.

$$4183.92+1528.8x-65000x^{2}=0$$

Subtract 4183.92 from both sides. Anything subtracted from zero gives its negation.

$$1528.8x-65000x^{2}=-4183.92$$

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.

$$-65000x^{2}+1528.8x=-4183.92$$

Divide both sides by -65000.

$$\frac{-65000x^{2}+1528.8x}{-65000}=-\frac{4183.92}{-65000}$$

Dividing by -65000 undoes the multiplication by -65000.

$$x^{2}+\frac{1528.8}{-65000}x=-\frac{4183.92}{-65000}$$

Divide 1528.8 by -65000.

$$x^{2}-0.02352x=-\frac{4183.92}{-65000}$$

Divide -4183.92 by -65000.

$$x^{2}-0.02352x=0.064368$$

Divide -0.02352, the coefficient of the x term, by 2 to get -0.01176. Then add the square of -0.01176 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0.02352x+\left(-0.01176\right)^{2}=0.064368+\left(-0.01176\right)^{2}$$

Square -0.01176 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0.02352x+0.0001382976=0.064368+0.0001382976$$

Add 0.064368 to 0.0001382976 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0.02352x+0.0001382976=0.0645062976$$

Factor x^{2}-0.02352x+0.0001382976. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.01176\right)^{2}=0.0645062976$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.01176\right)^{2}}=\sqrt{0.0645062976}$$

Simplify.

$$x-0.01176=\frac{3\sqrt{1119901}}{12500} x-0.01176=-\frac{3\sqrt{1119901}}{12500}$$

Add 0.01176 to both sides of the equation.

$$x=\frac{3\sqrt{1119901}+147}{12500} x=\frac{147-3\sqrt{1119901}}{12500}$$