Solve x^2-80x+1500

$${ x }^{ 2 } -80x+1500$$

Factor

$$\left(x-50\right)\left(x-30\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+1500. To find a and b, set up a system to be solved.

$$a+b=-80 ab=1\times 1500=1500$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 1500.

$$-1,-1500 -2,-750 -3,-500 -4,-375 -5,-300 -6,-250 -10,-150 -12,-125 -15,-100 -20,-75 -25,-60 -30,-50$$

Calculate the sum for each pair.

$$-1-1500=-1501 -2-750=-752 -3-500=-503 -4-375=-379 -5-300=-305 -6-250=-256 -10-150=-160 -12-125=-137 -15-100=-115 -20-75=-95 -25-60=-85 -30-50=-80$$

The solution is the pair that gives sum -80.

$$a=-50 b=-30$$

Rewrite x^{2}-80x+1500 as \left(x^{2}-50x\right)+\left(-30x+1500\right).

$$\left(x^{2}-50x\right)+\left(-30x+1500\right)$$

Factor out x in the first and -30 in the second group.

$$x\left(x-50\right)-30\left(x-50\right)$$

Factor out common term x-50 by using distributive property.

$$\left(x-50\right)\left(x-30\right)$$

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.

$$x^{2}-80x+1500=0$$

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.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 1500}}{2}$$

Square -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 1500}}{2}$$

Multiply -4 times 1500.

$$x=\frac{-\left(-80\right)±\sqrt{6400-6000}}{2}$$

Add 6400 to -6000.

$$x=\frac{-\left(-80\right)±\sqrt{400}}{2}$$

Take the square root of 400.

$$x=\frac{-\left(-80\right)±20}{2}$$

The opposite of -80 is 80.

$$x=\frac{80±20}{2}$$

Now solve the equation x=\frac{80±20}{2} when ± is plus. Add 80 to 20.

$$x=\frac{100}{2}$$

Divide 100 by 2.

$$x=50$$

Now solve the equation x=\frac{80±20}{2} when ± is minus. Subtract 20 from 80.

$$x=\frac{60}{2}$$

Divide 60 by 2.

$$x=30$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 50 for x_{1} and 30 for x_{2}.

$$x^{2}-80x+1500=\left(x-50\right)\left(x-30\right)$$

Evaluate

$$\left(x-50\right)\left(x-30\right)$$