Solve x^2-5x-6000

$${ x }^{ 2 } -5x-6000$$

Factor

$$\left(x-80\right)\left(x+75\right)$$

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.

$$a+b=-5 ab=1\left(-6000\right)=-6000$$

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 -5.

$$a=-80 b=75$$

Rewrite x^{2}-5x-6000 as \left(x^{2}-80x\right)+\left(75x-6000\right).

$$\left(x^{2}-80x\right)+\left(75x-6000\right)$$

Factor out x in the first and 75 in the second group.

$$x\left(x-80\right)+75\left(x-80\right)$$

Factor out common term x-80 by using distributive property.

$$\left(x-80\right)\left(x+75\right)$$

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.

$$x^{2}-5x-6000=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-6000\right)}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-6000\right)}}{2}$$

Multiply -4 times -6000.

$$x=\frac{-\left(-5\right)±\sqrt{25+24000}}{2}$$

Add 25 to 24000.

$$x=\frac{-\left(-5\right)±\sqrt{24025}}{2}$$

Take the square root of 24025.

$$x=\frac{-\left(-5\right)±155}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±155}{2}$$

Now solve the equation x=\frac{5±155}{2} when ± is plus. Add 5 to 155.

$$x=\frac{160}{2}$$

Divide 160 by 2.

$$x=80$$

Now solve the equation x=\frac{5±155}{2} when ± is minus. Subtract 155 from 5.

$$x=-\frac{150}{2}$$

Divide -150 by 2.

$$x=-75$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 80 for x_{1} and -75 for x_{2}.

$$x^{2}-5x-6000=\left(x-80\right)\left(x-\left(-75\right)\right)$$

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

$$x^{2}-5x-6000=\left(x-80\right)\left(x+75\right)$$

Evaluate

$$\left(x-80\right)\left(x+75\right)$$