Solve x^2-40x-6000

$$x ^ { 2 } - 40 x - 6000$$

Factor

$$\left(x-100\right)\left(x+60\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-6000. To find a and b, set up a system to be solved.

$$a+b=-40 ab=1\left(-6000\right)=-6000$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6000.

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

Calculate the sum for each pair.

$$1-6000=-5999 2-3000=-2998 3-2000=-1997 4-1500=-1496 5-1200=-1195 6-1000=-994 8-750=-742 10-600=-590 12-500=-488 15-400=-385 16-375=-359 20-300=-280 24-250=-226 25-240=-215 30-200=-170 40-150=-110 48-125=-77 50-120=-70 60-100=-40 75-80=-5$$

The solution is the pair that gives sum -40.

$$a=-100 b=60$$

Rewrite x^{2}-40x-6000 as \left(x^{2}-100x\right)+\left(60x-6000\right).

$$\left(x^{2}-100x\right)+\left(60x-6000\right)$$

Factor out x in the first and 60 in the second group.

$$x\left(x-100\right)+60\left(x-100\right)$$

Factor out common term x-100 by using distributive property.

$$\left(x-100\right)\left(x+60\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}-40x-6000=0$$

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.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\left(-6000\right)}}{2}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\left(-6000\right)}}{2}$$

Multiply -4 times -6000.

$$x=\frac{-\left(-40\right)±\sqrt{1600+24000}}{2}$$

Add 1600 to 24000.

$$x=\frac{-\left(-40\right)±\sqrt{25600}}{2}$$

Take the square root of 25600.

$$x=\frac{-\left(-40\right)±160}{2}$$

The opposite of -40 is 40.

$$x=\frac{40±160}{2}$$

Now solve the equation x=\frac{40±160}{2} when ± is plus. Add 40 to 160.

$$x=\frac{200}{2}$$

Divide 200 by 2.

$$x=100$$

Now solve the equation x=\frac{40±160}{2} when ± is minus. Subtract 160 from 40.

$$x=-\frac{120}{2}$$

Divide -120 by 2.

$$x=-60$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 100 for x_{1} and -60 for x_{2}.

$$x^{2}-40x-6000=\left(x-100\right)\left(x-\left(-60\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}-40x-6000=\left(x-100\right)\left(x+60\right)$$

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.

$$x ^ 2 -40x -6000 = 0$$

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

$$r + s = 40 rs = -6000$$

Two numbers r and s sum up to 40 exactly when the average of the two numbers is \frac{1}{2}*40 = 20. 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>

$$r = 20 - u s = 20 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -6000

$$(20 - u) (20 + u) = -6000$$

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

$$400 - u^2 = -6000$$

Simplify the expression by subtracting 400 on both sides

$$-u^2 = -6000-400 = -6400$$

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

$$u^2 = 6400 u = \pm\sqrt{6400} = \pm 80$$

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

$$r =20 - 80 = -60 s = 20 + 80 = 100$$

Evaluate

$$\left(x-100\right)\left(x+60\right)$$