Solve x^2+50x-30000=0

$$x ^ { 2 } + 50 x - 30000 = 0$$

Solve for x

$$x=-200$$

$$x=150$$

Steps Using Factoring

To solve the equation, factor x^{2}+50x-30000 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.

$$a+b=50 ab=-30000$$

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 -30000.

$$-1,30000 -2,15000 -3,10000 -4,7500 -5,6000 -6,5000 -8,3750 -10,3000 -12,2500 -15,2000 -16,1875 -20,1500 -24,1250 -25,1200 -30,1000 -40,750 -48,625 -50,600 -60,500 -75,400 -80,375 -100,300 -120,250 -125,240 -150,200$$

Calculate the sum for each pair.

$$-1+30000=29999 -2+15000=14998 -3+10000=9997 -4+7500=7496 -5+6000=5995 -6+5000=4994 -8+3750=3742 -10+3000=2990 -12+2500=2488 -15+2000=1985 -16+1875=1859 -20+1500=1480 -24+1250=1226 -25+1200=1175 -30+1000=970 -40+750=710 -48+625=577 -50+600=550 -60+500=440 -75+400=325 -80+375=295 -100+300=200 -120+250=130 -125+240=115 -150+200=50$$

The solution is the pair that gives sum 50.

$$a=-150 b=200$$

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

$$\left(x-150\right)\left(x+200\right)$$

To find equation solutions, solve x-150=0 and x+200=0.

$$x=150 x=-200$$

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 x^{2}+ax+bx-30000. To find a and b, set up a system to be solved.

$$a+b=50 ab=1\left(-30000\right)=-30000$$

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 -30000.

$$-1,30000 -2,15000 -3,10000 -4,7500 -5,6000 -6,5000 -8,3750 -10,3000 -12,2500 -15,2000 -16,1875 -20,1500 -24,1250 -25,1200 -30,1000 -40,750 -48,625 -50,600 -60,500 -75,400 -80,375 -100,300 -120,250 -125,240 -150,200$$

Calculate the sum for each pair.

$$-1+30000=29999 -2+15000=14998 -3+10000=9997 -4+7500=7496 -5+6000=5995 -6+5000=4994 -8+3750=3742 -10+3000=2990 -12+2500=2488 -15+2000=1985 -16+1875=1859 -20+1500=1480 -24+1250=1226 -25+1200=1175 -30+1000=970 -40+750=710 -48+625=577 -50+600=550 -60+500=440 -75+400=325 -80+375=295 -100+300=200 -120+250=130 -125+240=115 -150+200=50$$

The solution is the pair that gives sum 50.

$$a=-150 b=200$$

Rewrite x^{2}+50x-30000 as \left(x^{2}-150x\right)+\left(200x-30000\right).

$$\left(x^{2}-150x\right)+\left(200x-30000\right)$$

Factor out x in the first and 200 in the second group.

$$x\left(x-150\right)+200\left(x-150\right)$$

Factor out common term x-150 by using distributive property.

$$\left(x-150\right)\left(x+200\right)$$

To find equation solutions, solve x-150=0 and x+200=0.

$$x=150 x=-200$$

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.

$$x^{2}+50x-30000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 50 for b, and -30000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\left(-30000\right)}}{2}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\left(-30000\right)}}{2}$$

Multiply -4 times -30000.

$$x=\frac{-50±\sqrt{2500+120000}}{2}$$

Add 2500 to 120000.

$$x=\frac{-50±\sqrt{122500}}{2}$$

Take the square root of 122500.

$$x=\frac{-50±350}{2}$$

Now solve the equation x=\frac{-50±350}{2} when ± is plus. Add -50 to 350.

$$x=\frac{300}{2}$$

Divide 300 by 2.

$$x=150$$

Now solve the equation x=\frac{-50±350}{2} when ± is minus. Subtract 350 from -50.

$$x=-\frac{400}{2}$$

Divide -400 by 2.

$$x=-200$$

The equation is now solved.

$$x=150 x=-200$$

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.

$$x^{2}+50x-30000=0$$

Add 30000 to both sides of the equation.

$$x^{2}+50x-30000-\left(-30000\right)=-\left(-30000\right)$$

Subtracting -30000 from itself leaves 0.

$$x^{2}+50x=-\left(-30000\right)$$

Subtract -30000 from 0.

$$x^{2}+50x=30000$$

Divide 50, the coefficient of the x term, by 2 to get 25. Then add the square of 25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+50x+25^{2}=30000+25^{2}$$

Square 25.

$$x^{2}+50x+625=30000+625$$

Add 30000 to 625.

$$x^{2}+50x+625=30625$$

Factor x^{2}+50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+25\right)^{2}=30625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+25\right)^{2}}=\sqrt{30625}$$

Simplify.

$$x+25=175 x+25=-175$$

Subtract 25 from both sides of the equation.

$$x=150 x=-200$$

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.

$$x ^ 2 +50x -30000 = 0$$

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

$$r + s = -50 rs = -30000$$

Two numbers r and s sum up to -50 exactly when the average of the two numbers is \frac{1}{2}*-50 = -25. 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>

$$r = -25 - u s = -25 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -30000

$$(-25 - u) (-25 + u) = -30000$$

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

$$625 - u^2 = -30000$$

Simplify the expression by subtracting 625 on both sides

$$-u^2 = -30000-625 = -30625$$

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

$$u^2 = 30625 u = \pm\sqrt{30625} = \pm 175$$

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

$$r =-25 - 175 = -200 s = -25 + 175 = 150$$