Solve -x^2+210x-8000

$$- x ^ { 2 } + 210 x - 8000$$

Factor

$$\left(50-x\right)\left(x-160\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-8000. To find a and b, set up a system to be solved.

$$a+b=210 ab=-\left(-8000\right)=8000$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 8000.

$$1,8000 2,4000 4,2000 5,1600 8,1000 10,800 16,500 20,400 25,320 32,250 40,200 50,160 64,125 80,100$$

Calculate the sum for each pair.

$$1+8000=8001 2+4000=4002 4+2000=2004 5+1600=1605 8+1000=1008 10+800=810 16+500=516 20+400=420 25+320=345 32+250=282 40+200=240 50+160=210 64+125=189 80+100=180$$

The solution is the pair that gives sum 210.

$$a=160 b=50$$

Rewrite -x^{2}+210x-8000 as \left(-x^{2}+160x\right)+\left(50x-8000\right).

$$\left(-x^{2}+160x\right)+\left(50x-8000\right)$$

Factor out -x in the first and 50 in the second group.

$$-x\left(x-160\right)+50\left(x-160\right)$$

Factor out common term x-160 by using distributive property.

$$\left(x-160\right)\left(-x+50\right)$$

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.

$$-x^{2}+210x-8000=0$$

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.

$$x=\frac{-210±\sqrt{210^{2}-4\left(-1\right)\left(-8000\right)}}{2\left(-1\right)}$$

Square 210.

$$x=\frac{-210±\sqrt{44100-4\left(-1\right)\left(-8000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-210±\sqrt{44100+4\left(-8000\right)}}{2\left(-1\right)}$$

Multiply 4 times -8000.

$$x=\frac{-210±\sqrt{44100-32000}}{2\left(-1\right)}$$

Add 44100 to -32000.

$$x=\frac{-210±\sqrt{12100}}{2\left(-1\right)}$$

Take the square root of 12100.

$$x=\frac{-210±110}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-210±110}{-2}$$

Now solve the equation x=\frac{-210±110}{-2} when ± is plus. Add -210 to 110.

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

Divide -100 by -2.

$$x=50$$

Now solve the equation x=\frac{-210±110}{-2} when ± is minus. Subtract 110 from -210.

$$x=-\frac{320}{-2}$$

Divide -320 by -2.

$$x=160$$

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 160 for x_{2}.

$$-x^{2}+210x-8000=-\left(x-50\right)\left(x-160\right)$$

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.

$$x ^ 2 -210x +8000 = 0$$

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

$$r + s = 210 rs = 8000$$

Two numbers r and s sum up to 210 exactly when the average of the two numbers is \frac{1}{2}*210 = 105. 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>

$$r = 105 - u s = 105 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 8000

$$(105 - u) (105 + u) = 8000$$

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

$$11025 - u^2 = 8000$$

Simplify the expression by subtracting 11025 on both sides

$$-u^2 = 8000-11025 = -3025$$

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

$$u^2 = 3025 u = \pm\sqrt{3025} = \pm 55$$

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

$$r =105 - 55 = 50 s = 105 + 55 = 160$$

Evaluate

$$\left(50-x\right)\left(x-160\right)$$