Solve 100x^2+200x-189=0

$$100 x ^ { 2 } + 200 x - 189 = 0$$

Solve for x

$$x = -\frac{27}{10} = -2\frac{7}{10} = -2.7$$

$$x=\frac{7}{10}=0.7$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 100x^{2}+ax+bx-189. To find a and b, set up a system to be solved.

$$a+b=200 ab=100\left(-189\right)=-18900$$

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 -18900.

$$-1,18900 -2,9450 -3,6300 -4,4725 -5,3780 -6,3150 -7,2700 -9,2100 -10,1890 -12,1575 -14,1350 -15,1260 -18,1050 -20,945 -21,900 -25,756 -27,700 -28,675 -30,630 -35,540 -36,525 -42,450 -45,420 -50,378 -54,350 -60,315 -63,300 -70,270 -75,252 -84,225 -90,210 -100,189 -105,180 -108,175 -126,150 -135,140$$

Calculate the sum for each pair.

$$-1+18900=18899 -2+9450=9448 -3+6300=6297 -4+4725=4721 -5+3780=3775 -6+3150=3144 -7+2700=2693 -9+2100=2091 -10+1890=1880 -12+1575=1563 -14+1350=1336 -15+1260=1245 -18+1050=1032 -20+945=925 -21+900=879 -25+756=731 -27+700=673 -28+675=647 -30+630=600 -35+540=505 -36+525=489 -42+450=408 -45+420=375 -50+378=328 -54+350=296 -60+315=255 -63+300=237 -70+270=200 -75+252=177 -84+225=141 -90+210=120 -100+189=89 -105+180=75 -108+175=67 -126+150=24 -135+140=5$$

The solution is the pair that gives sum 200.

$$a=-70 b=270$$

Rewrite 100x^{2}+200x-189 as \left(100x^{2}-70x\right)+\left(270x-189\right).

$$\left(100x^{2}-70x\right)+\left(270x-189\right)$$

Factor out 10x in the first and 27 in the second group.

$$10x\left(10x-7\right)+27\left(10x-7\right)$$

Factor out common term 10x-7 by using distributive property.

$$\left(10x-7\right)\left(10x+27\right)$$

To find equation solutions, solve 10x-7=0 and 10x+27=0.

$$x=\frac{7}{10} x=-\frac{27}{10}$$

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.

$$100x^{2}+200x-189=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 200 for b, and -189 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\times 100\left(-189\right)}}{2\times 100}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\times 100\left(-189\right)}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{-200±\sqrt{40000-400\left(-189\right)}}{2\times 100}$$

Multiply -400 times -189.

$$x=\frac{-200±\sqrt{40000+75600}}{2\times 100}$$

Add 40000 to 75600.

$$x=\frac{-200±\sqrt{115600}}{2\times 100}$$

Take the square root of 115600.

$$x=\frac{-200±340}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{-200±340}{200}$$

Now solve the equation x=\frac{-200±340}{200} when ± is plus. Add -200 to 340.

$$x=\frac{140}{200}$$

Reduce the fraction \frac{140}{200} to lowest terms by extracting and canceling out 20.

$$x=\frac{7}{10}$$

Now solve the equation x=\frac{-200±340}{200} when ± is minus. Subtract 340 from -200.

$$x=-\frac{540}{200}$$

Reduce the fraction \frac{-540}{200} to lowest terms by extracting and canceling out 20.

$$x=-\frac{27}{10}$$

The equation is now solved.

$$x=\frac{7}{10} x=-\frac{27}{10}$$

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.

$$100x^{2}+200x-189=0$$

Add 189 to both sides of the equation.

$$100x^{2}+200x-189-\left(-189\right)=-\left(-189\right)$$

Subtracting -189 from itself leaves 0.

$$100x^{2}+200x=-\left(-189\right)$$

Subtract -189 from 0.

$$100x^{2}+200x=189$$

Divide both sides by 100.

$$\frac{100x^{2}+200x}{100}=\frac{189}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$x^{2}+\frac{200}{100}x=\frac{189}{100}$$

Divide 200 by 100.

$$x^{2}+2x=\frac{189}{100}$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=\frac{189}{100}+1^{2}$$

Square 1.

$$x^{2}+2x+1=\frac{189}{100}+1$$

Add \frac{189}{100} to 1.

$$x^{2}+2x+1=\frac{289}{100}$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=\frac{289}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{289}{100}}$$

Simplify.

$$x+1=\frac{17}{10} x+1=-\frac{17}{10}$$

Subtract 1 from both sides of the equation.

$$x=\frac{7}{10} x=-\frac{27}{10}$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 100

$$x ^ 2 +2x -\frac{189}{100} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -2 rs = -\frac{189}{100}$$

Two numbers r and s sum up to -2 exactly when the average of the two numbers is \frac{1}{2}*-2 = -1. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -1 - u s = -1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{189}{100}

$$(-1 - u) (-1 + u) = -\frac{189}{100}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$1 - u^2 = -\frac{189}{100}$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -\frac{189}{100}-1 = -\frac{289}{100}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{289}{100} u = \pm\sqrt{\frac{289}{100}} = \pm \frac{17}{10}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-1 - \frac{17}{10} = -2.700 s = -1 + \frac{17}{10} = 0.700$$