Solve 34*10^{3}*9.8x+1/2*34*10^3*100/34^2*2*9.8=2*100*9.8x+2.65*10^4x^2

$$34 \times 10 ^ { 3 } \times 9.8 x + \frac { 1 } { 2 } \times 34 \times 10 ^ { 3 } \times \frac { 100 } { 34 ^ { 2 } } \times 2 \times 9.8 = 2 \times 100 \times 9.8 x + 2.65 \times 10 ^ { 4 } x ^ { 2 }$$

Solve for x

$$x=\frac{7\sqrt{415714821}}{22525}+\frac{8281}{1325}\approx 12.586042211$$

$$x=-\frac{7\sqrt{415714821}}{22525}+\frac{8281}{1325}\approx -0.08641957$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 3 and get 1000.

$$34\times 1000\times 9.8x+\frac{1}{2}\times 34\times 10^{3}\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 34 and 1000 to get 34000.

$$34000\times 9.8x+\frac{1}{2}\times 34\times 10^{3}\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 34000 and 9.8 to get 333200.

$$333200x+\frac{1}{2}\times 34\times 10^{3}\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply \frac{1}{2} and 34 to get 17.

$$333200x+17\times 10^{3}\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Calculate 10 to the power of 3 and get 1000.

$$333200x+17\times 1000\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 17 and 1000 to get 17000.

$$333200x+17000\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Calculate 34 to the power of 2 and get 1156.

$$333200x+17000\times \frac{100}{1156}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Reduce the fraction \frac{100}{1156} to lowest terms by extracting and canceling out 4.

$$333200x+17000\times \frac{25}{289}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 17000 and \frac{25}{289} to get \frac{25000}{17}.

$$333200x+\frac{25000}{17}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply \frac{25000}{17} and 2 to get \frac{50000}{17}.

$$333200x+\frac{50000}{17}\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply \frac{50000}{17} and 9.8 to get \frac{490000}{17}.

$$333200x+\frac{490000}{17}=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 2 and 100 to get 200.

$$333200x+\frac{490000}{17}=200\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 200 and 9.8 to get 1960.

$$333200x+\frac{490000}{17}=1960x+2.65\times 10^{4}x^{2}$$

Calculate 10 to the power of 4 and get 10000.

$$333200x+\frac{490000}{17}=1960x+2.65\times 10000x^{2}$$

Multiply 2.65 and 10000 to get 26500.

$$333200x+\frac{490000}{17}=1960x+26500x^{2}$$

Subtract 1960x from both sides.

$$333200x+\frac{490000}{17}-1960x=26500x^{2}$$

Combine 333200x and -1960x to get 331240x.

$$331240x+\frac{490000}{17}=26500x^{2}$$

Subtract 26500x^{2} from both sides.

$$331240x+\frac{490000}{17}-26500x^{2}=0$$

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.

$$-26500x^{2}+331240x+\frac{490000}{17}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -26500 for a, 331240 for b, and \frac{490000}{17} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-331240±\sqrt{331240^{2}-4\left(-26500\right)\times \frac{490000}{17}}}{2\left(-26500\right)}$$

Square 331240.

$$x=\frac{-331240±\sqrt{109719937600-4\left(-26500\right)\times \frac{490000}{17}}}{2\left(-26500\right)}$$

Multiply -4 times -26500.

$$x=\frac{-331240±\sqrt{109719937600+106000\times \frac{490000}{17}}}{2\left(-26500\right)}$$

Multiply 106000 times \frac{490000}{17}.

$$x=\frac{-331240±\sqrt{109719937600+\frac{51940000000}{17}}}{2\left(-26500\right)}$$

Add 109719937600 to \frac{51940000000}{17}.

$$x=\frac{-331240±\sqrt{\frac{1917178939200}{17}}}{2\left(-26500\right)}$$

Take the square root of \frac{1917178939200}{17}.

$$x=\frac{-331240±\frac{280\sqrt{415714821}}{17}}{2\left(-26500\right)}$$

Multiply 2 times -26500.

$$x=\frac{-331240±\frac{280\sqrt{415714821}}{17}}{-53000}$$

Now solve the equation x=\frac{-331240±\frac{280\sqrt{415714821}}{17}}{-53000} when ± is plus. Add -331240 to \frac{280\sqrt{415714821}}{17}.

$$x=\frac{\frac{280\sqrt{415714821}}{17}-331240}{-53000}$$

Divide -331240+\frac{280\sqrt{415714821}}{17} by -53000.

$$x=-\frac{7\sqrt{415714821}}{22525}+\frac{8281}{1325}$$

Now solve the equation x=\frac{-331240±\frac{280\sqrt{415714821}}{17}}{-53000} when ± is minus. Subtract \frac{280\sqrt{415714821}}{17} from -331240.

$$x=\frac{-\frac{280\sqrt{415714821}}{17}-331240}{-53000}$$

Divide -331240-\frac{280\sqrt{415714821}}{17} by -53000.

$$x=\frac{7\sqrt{415714821}}{22525}+\frac{8281}{1325}$$

The equation is now solved.

$$x=-\frac{7\sqrt{415714821}}{22525}+\frac{8281}{1325} x=\frac{7\sqrt{415714821}}{22525}+\frac{8281}{1325}$$

Steps for Completing the Square

Calculate 10 to the power of 3 and get 1000.

$$34\times 1000\times 9.8x+\frac{1}{2}\times 34\times 10^{3}\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 34 and 1000 to get 34000.

$$34000\times 9.8x+\frac{1}{2}\times 34\times 10^{3}\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 34000 and 9.8 to get 333200.

$$333200x+\frac{1}{2}\times 34\times 10^{3}\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply \frac{1}{2} and 34 to get 17.

$$333200x+17\times 10^{3}\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Calculate 10 to the power of 3 and get 1000.

$$333200x+17\times 1000\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 17 and 1000 to get 17000.

$$333200x+17000\times \frac{100}{34^{2}}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Calculate 34 to the power of 2 and get 1156.

$$333200x+17000\times \frac{100}{1156}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Reduce the fraction \frac{100}{1156} to lowest terms by extracting and canceling out 4.

$$333200x+17000\times \frac{25}{289}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 17000 and \frac{25}{289} to get \frac{25000}{17}.

$$333200x+\frac{25000}{17}\times 2\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply \frac{25000}{17} and 2 to get \frac{50000}{17}.

$$333200x+\frac{50000}{17}\times 9.8=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply \frac{50000}{17} and 9.8 to get \frac{490000}{17}.

$$333200x+\frac{490000}{17}=2\times 100\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 2 and 100 to get 200.

$$333200x+\frac{490000}{17}=200\times 9.8x+2.65\times 10^{4}x^{2}$$

Multiply 200 and 9.8 to get 1960.

$$333200x+\frac{490000}{17}=1960x+2.65\times 10^{4}x^{2}$$

Calculate 10 to the power of 4 and get 10000.

$$333200x+\frac{490000}{17}=1960x+2.65\times 10000x^{2}$$

Multiply 2.65 and 10000 to get 26500.

$$333200x+\frac{490000}{17}=1960x+26500x^{2}$$

Subtract 1960x from both sides.

$$333200x+\frac{490000}{17}-1960x=26500x^{2}$$

Combine 333200x and -1960x to get 331240x.

$$331240x+\frac{490000}{17}=26500x^{2}$$

Subtract 26500x^{2} from both sides.

$$331240x+\frac{490000}{17}-26500x^{2}=0$$

Subtract \frac{490000}{17} from both sides. Anything subtracted from zero gives its negation.

$$331240x-26500x^{2}=-\frac{490000}{17}$$

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.

$$-26500x^{2}+331240x=-\frac{490000}{17}$$

Divide both sides by -26500.

$$\frac{-26500x^{2}+331240x}{-26500}=-\frac{\frac{490000}{17}}{-26500}$$

Dividing by -26500 undoes the multiplication by -26500.

$$x^{2}+\frac{331240}{-26500}x=-\frac{\frac{490000}{17}}{-26500}$$

Reduce the fraction \frac{331240}{-26500} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{16562}{1325}x=-\frac{\frac{490000}{17}}{-26500}$$

Divide -\frac{490000}{17} by -26500.

$$x^{2}-\frac{16562}{1325}x=\frac{980}{901}$$

Divide -\frac{16562}{1325}, the coefficient of the x term, by 2 to get -\frac{8281}{1325}. Then add the square of -\frac{8281}{1325} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16562}{1325}x+\left(-\frac{8281}{1325}\right)^{2}=\frac{980}{901}+\left(-\frac{8281}{1325}\right)^{2}$$

Square -\frac{8281}{1325} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16562}{1325}x+\frac{68574961}{1755625}=\frac{980}{901}+\frac{68574961}{1755625}$$

Add \frac{980}{901} to \frac{68574961}{1755625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16562}{1325}x+\frac{68574961}{1755625}=\frac{1198236837}{29845625}$$

Factor x^{2}-\frac{16562}{1325}x+\frac{68574961}{1755625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8281}{1325}\right)^{2}=\frac{1198236837}{29845625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8281}{1325}\right)^{2}}=\sqrt{\frac{1198236837}{29845625}}$$

Simplify.

$$x-\frac{8281}{1325}=\frac{7\sqrt{415714821}}{22525} x-\frac{8281}{1325}=-\frac{7\sqrt{415714821}}{22525}$$

Add \frac{8281}{1325} to both sides of the equation.

$$x=\frac{7\sqrt{415714821}}{22525}+\frac{8281}{1325} x=-\frac{7\sqrt{415714821}}{22525}+\frac{8281}{1325}$$