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$\exponential{x}{2} + 30 x - 18000 = 0 $
Solve for x
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a+b=30 ab=-18000
To solve the equation, factor x^{2}+30x-18000 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.
-1,18000 -2,9000 -3,6000 -4,4500 -5,3600 -6,3000 -8,2250 -9,2000 -10,1800 -12,1500 -15,1200 -16,1125 -18,1000 -20,900 -24,750 -25,720 -30,600 -36,500 -40,450 -45,400 -48,375 -50,360 -60,300 -72,250 -75,240 -80,225 -90,200 -100,180 -120,150 -125,144
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 -18000.
-1+18000=17999 -2+9000=8998 -3+6000=5997 -4+4500=4496 -5+3600=3595 -6+3000=2994 -8+2250=2242 -9+2000=1991 -10+1800=1790 -12+1500=1488 -15+1200=1185 -16+1125=1109 -18+1000=982 -20+900=880 -24+750=726 -25+720=695 -30+600=570 -36+500=464 -40+450=410 -45+400=355 -48+375=327 -50+360=310 -60+300=240 -72+250=178 -75+240=165 -80+225=145 -90+200=110 -100+180=80 -120+150=30 -125+144=19
Calculate the sum for each pair.
a=-120 b=150
The solution is the pair that gives sum 30.
\left(x-120\right)\left(x+150\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=120 x=-150
To find equation solutions, solve x-120=0 and x+150=0.
a+b=30 ab=1\left(-18000\right)=-18000
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-18000. To find a and b, set up a system to be solved.
-1,18000 -2,9000 -3,6000 -4,4500 -5,3600 -6,3000 -8,2250 -9,2000 -10,1800 -12,1500 -15,1200 -16,1125 -18,1000 -20,900 -24,750 -25,720 -30,600 -36,500 -40,450 -45,400 -48,375 -50,360 -60,300 -72,250 -75,240 -80,225 -90,200 -100,180 -120,150 -125,144
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 -18000.
-1+18000=17999 -2+9000=8998 -3+6000=5997 -4+4500=4496 -5+3600=3595 -6+3000=2994 -8+2250=2242 -9+2000=1991 -10+1800=1790 -12+1500=1488 -15+1200=1185 -16+1125=1109 -18+1000=982 -20+900=880 -24+750=726 -25+720=695 -30+600=570 -36+500=464 -40+450=410 -45+400=355 -48+375=327 -50+360=310 -60+300=240 -72+250=178 -75+240=165 -80+225=145 -90+200=110 -100+180=80 -120+150=30 -125+144=19
Calculate the sum for each pair.
a=-120 b=150
The solution is the pair that gives sum 30.
\left(x^{2}-120x\right)+\left(150x-18000\right)
Rewrite x^{2}+30x-18000 as \left(x^{2}-120x\right)+\left(150x-18000\right).
x\left(x-120\right)+150\left(x-120\right)
Factor out x in the first and 150 in the second group.
\left(x-120\right)\left(x+150\right)
Factor out common term x-120 by using distributive property.
x=120 x=-150
To find equation solutions, solve x-120=0 and x+150=0.
x^{2}+30x-18000=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.
x=\frac{-30±\sqrt{30^{2}-4\left(-18000\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 30 for b, and -18000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-30±\sqrt{900-4\left(-18000\right)}}{2}
Square 30.
x=\frac{-30±\sqrt{900+72000}}{2}
Multiply -4 times -18000.
x=\frac{-30±\sqrt{72900}}{2}
Add 900 to 72000.
x=\frac{-30±270}{2}
Take the square root of 72900.
x=\frac{240}{2}
Now solve the equation x=\frac{-30±270}{2} when ± is plus. Add -30 to 270.
x=120
Divide 240 by 2.
x=-\frac{300}{2}
Now solve the equation x=\frac{-30±270}{2} when ± is minus. Subtract 270 from -30.
x=-150
Divide -300 by 2.
x=120 x=-150
The equation is now solved.
x^{2}+30x-18000=0
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}+30x-18000-\left(-18000\right)=-\left(-18000\right)
Add 18000 to both sides of the equation.
x^{2}+30x=-\left(-18000\right)
Subtracting -18000 from itself leaves 0.
x^{2}+30x=18000
Subtract -18000 from 0.
x^{2}+30x+15^{2}=18000+15^{2}
Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+30x+225=18000+225
Square 15.
x^{2}+30x+225=18225
Add 18000 to 225.
\left(x+15\right)^{2}=18225
Factor x^{2}+30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+15\right)^{2}}=\sqrt{18225}
Take the square root of both sides of the equation.
x+15=135 x+15=-135
Simplify.
x=120 x=-150
Subtract 15 from both sides of the equation.
x ^ 2 +30x -18000 = 0
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.
r + s = -30 rs = -18000
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 = -15 - u s = -15 + u
Two numbers r and s sum up to -30 exactly when the average of the two numbers is \frac{1}{2}*-30 = -15. 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>
(-15 - u) (-15 + u) = -18000
To solve for unknown quantity u, substitute these in the product equation rs = -18000
225 - u^2 = -18000
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -18000-225 = -18225
Simplify the expression by subtracting 225 on both sides
u^2 = 18225 u = \pm\sqrt{18225} = \pm 135
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-15 - 135 = -150 s = -15 + 135 = 120
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.