Solve 3830.7=0.2*15*9.81*(100+x)+0.5*2.5*x^2

$$3830.7 = 0.2 \times 15 \times 9.81 \times ( 100 + x ) + 0.5 \times 2.5 \times x ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{53046249} - 2943}{250} \approx 17.361142364$$

$$x=\frac{-\sqrt{53046249}-2943}{250}\approx -40.905142364$$

Steps Using the Quadratic Formula

Multiply 0.2 and 15 to get 3.

$$3830.7=3\times 9.81\left(100+x\right)+0.5\times 2.5x^{2}$$

Multiply 3 and 9.81 to get 29.43.

$$3830.7=29.43\left(100+x\right)+0.5\times 2.5x^{2}$$

Use the distributive property to multiply 29.43 by 100+x.

$$3830.7=2943+29.43x+0.5\times 2.5x^{2}$$

Multiply 0.5 and 2.5 to get 1.25.

$$3830.7=2943+29.43x+1.25x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$2943+29.43x+1.25x^{2}=3830.7$$

Subtract 3830.7 from both sides.

$$2943+29.43x+1.25x^{2}-3830.7=0$$

Subtract 3830.7 from 2943 to get -887.7.

$$-887.7+29.43x+1.25x^{2}=0$$

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.

$$1.25x^{2}+29.43x-887.7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.25 for a, 29.43 for b, and -887.7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-29.43±\sqrt{29.43^{2}-4\times 1.25\left(-887.7\right)}}{2\times 1.25}$$

Square 29.43 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-29.43±\sqrt{866.1249-4\times 1.25\left(-887.7\right)}}{2\times 1.25}$$

Multiply -4 times 1.25.

$$x=\frac{-29.43±\sqrt{866.1249-5\left(-887.7\right)}}{2\times 1.25}$$

Multiply -5 times -887.7.

$$x=\frac{-29.43±\sqrt{866.1249+4438.5}}{2\times 1.25}$$

Add 866.1249 to 4438.5 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-29.43±\sqrt{5304.6249}}{2\times 1.25}$$

Take the square root of 5304.6249.

$$x=\frac{-29.43±\frac{\sqrt{53046249}}{100}}{2\times 1.25}$$

Multiply 2 times 1.25.

$$x=\frac{-29.43±\frac{\sqrt{53046249}}{100}}{2.5}$$

Now solve the equation x=\frac{-29.43±\frac{\sqrt{53046249}}{100}}{2.5} when ± is plus. Add -29.43 to \frac{\sqrt{53046249}}{100}.

$$x=\frac{\sqrt{53046249}-2943}{2.5\times 100}$$

Divide \frac{-2943+\sqrt{53046249}}{100} by 2.5 by multiplying \frac{-2943+\sqrt{53046249}}{100} by the reciprocal of 2.5.

$$x=\frac{\sqrt{53046249}-2943}{250}$$

Now solve the equation x=\frac{-29.43±\frac{\sqrt{53046249}}{100}}{2.5} when ± is minus. Subtract \frac{\sqrt{53046249}}{100} from -29.43.

$$x=\frac{-\sqrt{53046249}-2943}{2.5\times 100}$$

Divide \frac{-2943-\sqrt{53046249}}{100} by 2.5 by multiplying \frac{-2943-\sqrt{53046249}}{100} by the reciprocal of 2.5.

$$x=\frac{-\sqrt{53046249}-2943}{250}$$

The equation is now solved.

$$x=\frac{\sqrt{53046249}-2943}{250} x=\frac{-\sqrt{53046249}-2943}{250}$$

Steps for Completing the Square

Multiply 0.2 and 15 to get 3.

$$3830.7=3\times 9.81\left(100+x\right)+0.5\times 2.5x^{2}$$

Multiply 3 and 9.81 to get 29.43.

$$3830.7=29.43\left(100+x\right)+0.5\times 2.5x^{2}$$

Use the distributive property to multiply 29.43 by 100+x.

$$3830.7=2943+29.43x+0.5\times 2.5x^{2}$$

Multiply 0.5 and 2.5 to get 1.25.

$$3830.7=2943+29.43x+1.25x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$2943+29.43x+1.25x^{2}=3830.7$$

Subtract 2943 from both sides.

$$29.43x+1.25x^{2}=3830.7-2943$$

Subtract 2943 from 3830.7 to get 887.7.

$$29.43x+1.25x^{2}=887.7$$

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.

$$1.25x^{2}+29.43x=887.7$$

Divide both sides of the equation by 1.25, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.25x^{2}+29.43x}{1.25}=\frac{887.7}{1.25}$$

Dividing by 1.25 undoes the multiplication by 1.25.

$$x^{2}+\frac{29.43}{1.25}x=\frac{887.7}{1.25}$$

Divide 29.43 by 1.25 by multiplying 29.43 by the reciprocal of 1.25.

$$x^{2}+23.544x=\frac{887.7}{1.25}$$

Divide 887.7 by 1.25 by multiplying 887.7 by the reciprocal of 1.25.

$$x^{2}+23.544x=710.16$$

Divide 23.544, the coefficient of the x term, by 2 to get 11.772. Then add the square of 11.772 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+23.544x+11.772^{2}=710.16+11.772^{2}$$

Square 11.772 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+23.544x+138.579984=710.16+138.579984$$

Add 710.16 to 138.579984 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+23.544x+138.579984=848.739984$$

Factor x^{2}+23.544x+138.579984. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+11.772\right)^{2}=848.739984$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+11.772\right)^{2}}=\sqrt{848.739984}$$

Simplify.

$$x+11.772=\frac{\sqrt{53046249}}{250} x+11.772=-\frac{\sqrt{53046249}}{250}$$

Subtract 11.772 from both sides of the equation.

$$x=\frac{\sqrt{53046249}-2943}{250} x=\frac{-\sqrt{53046249}-2943}{250}$$