Solve 200x+x^2=2100

$$200 x + x ^ { 2 } = 2100$$

Solve for x

$$x=-210$$

$$x=10$$

Steps Using Factoring

Subtract 2100 from both sides.

$$200x+x^{2}-2100=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+200x-2100=0$$

To solve the equation, factor x^{2}+200x-2100 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=200 ab=-2100$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2100.

$$-1,2100 -2,1050 -3,700 -4,525 -5,420 -6,350 -7,300 -10,210 -12,175 -14,150 -15,140 -20,105 -21,100 -25,84 -28,75 -30,70 -35,60 -42,50$$

Calculate the sum for each pair.

$$-1+2100=2099 -2+1050=1048 -3+700=697 -4+525=521 -5+420=415 -6+350=344 -7+300=293 -10+210=200 -12+175=163 -14+150=136 -15+140=125 -20+105=85 -21+100=79 -25+84=59 -28+75=47 -30+70=40 -35+60=25 -42+50=8$$

The solution is the pair that gives sum 200.

$$a=-10 b=210$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-10\right)\left(x+210\right)$$

To find equation solutions, solve x-10=0 and x+210=0.

$$x=10 x=-210$$

Steps Using Factoring By Grouping

Subtract 2100 from both sides.

$$200x+x^{2}-2100=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+200x-2100=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-2100. To find a and b, set up a system to be solved.

$$a+b=200 ab=1\left(-2100\right)=-2100$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2100.

$$-1,2100 -2,1050 -3,700 -4,525 -5,420 -6,350 -7,300 -10,210 -12,175 -14,150 -15,140 -20,105 -21,100 -25,84 -28,75 -30,70 -35,60 -42,50$$

Calculate the sum for each pair.

$$-1+2100=2099 -2+1050=1048 -3+700=697 -4+525=521 -5+420=415 -6+350=344 -7+300=293 -10+210=200 -12+175=163 -14+150=136 -15+140=125 -20+105=85 -21+100=79 -25+84=59 -28+75=47 -30+70=40 -35+60=25 -42+50=8$$

The solution is the pair that gives sum 200.

$$a=-10 b=210$$

Rewrite x^{2}+200x-2100 as \left(x^{2}-10x\right)+\left(210x-2100\right).

$$\left(x^{2}-10x\right)+\left(210x-2100\right)$$

Factor out x in the first and 210 in the second group.

$$x\left(x-10\right)+210\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(x+210\right)$$

To find equation solutions, solve x-10=0 and x+210=0.

$$x=10 x=-210$$

Steps Using the Quadratic Formula

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.

$$x^{2}+200x=2100$$

Subtract 2100 from both sides of the equation.

$$x^{2}+200x-2100=2100-2100$$

Subtracting 2100 from itself leaves 0.

$$x^{2}+200x-2100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 200 for b, and -2100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-2100\right)}}{2}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-2100\right)}}{2}$$

Multiply -4 times -2100.

$$x=\frac{-200±\sqrt{40000+8400}}{2}$$

Add 40000 to 8400.

$$x=\frac{-200±\sqrt{48400}}{2}$$

Take the square root of 48400.

$$x=\frac{-200±220}{2}$$

Now solve the equation x=\frac{-200±220}{2} when ± is plus. Add -200 to 220.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{-200±220}{2} when ± is minus. Subtract 220 from -200.

$$x=-\frac{420}{2}$$

Divide -420 by 2.

$$x=-210$$

The equation is now solved.

$$x=10 x=-210$$

Steps for Completing the Square

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.

$$x^{2}+200x=2100$$

Divide 200, the coefficient of the x term, by 2 to get 100. Then add the square of 100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+200x+100^{2}=2100+100^{2}$$

Square 100.

$$x^{2}+200x+10000=2100+10000$$

Add 2100 to 10000.

$$x^{2}+200x+10000=12100$$

Factor x^{2}+200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+100\right)^{2}=12100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+100\right)^{2}}=\sqrt{12100}$$

Simplify.

$$x+100=110 x+100=-110$$

Subtract 100 from both sides of the equation.

$$x=10 x=-210$$