Solve -10x^2+400x+45000

$$- 10 x ^ { 2 } + 400 x + 45000$$

Factor

$$-10\left(x-90\right)\left(x+50\right)$$

Solution Steps

Factor out 10.

$$10\left(-x^{2}+40x+4500\right)$$

Consider -x^{2}+40x+4500. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+4500. To find a and b, set up a system to be solved.

$$a+b=40 ab=-4500=-4500$$

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 -4500.

$$-1,4500 -2,2250 -3,1500 -4,1125 -5,900 -6,750 -9,500 -10,450 -12,375 -15,300 -18,250 -20,225 -25,180 -30,150 -36,125 -45,100 -50,90 -60,75$$

Calculate the sum for each pair.

$$-1+4500=4499 -2+2250=2248 -3+1500=1497 -4+1125=1121 -5+900=895 -6+750=744 -9+500=491 -10+450=440 -12+375=363 -15+300=285 -18+250=232 -20+225=205 -25+180=155 -30+150=120 -36+125=89 -45+100=55 -50+90=40 -60+75=15$$

The solution is the pair that gives sum 40.

$$a=90 b=-50$$

Rewrite -x^{2}+40x+4500 as \left(-x^{2}+90x\right)+\left(-50x+4500\right).

$$\left(-x^{2}+90x\right)+\left(-50x+4500\right)$$

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

$$-x\left(x-90\right)-50\left(x-90\right)$$

Factor out common term x-90 by using distributive property.

$$\left(x-90\right)\left(-x-50\right)$$

Rewrite the complete factored expression.

$$10\left(x-90\right)\left(-x-50\right)$$

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.

$$-10x^{2}+400x+45000=0$$

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.

$$x=\frac{-400±\sqrt{400^{2}-4\left(-10\right)\times 45000}}{2\left(-10\right)}$$

Square 400.

$$x=\frac{-400±\sqrt{160000-4\left(-10\right)\times 45000}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-400±\sqrt{160000+40\times 45000}}{2\left(-10\right)}$$

Multiply 40 times 45000.

$$x=\frac{-400±\sqrt{160000+1800000}}{2\left(-10\right)}$$

Add 160000 to 1800000.

$$x=\frac{-400±\sqrt{1960000}}{2\left(-10\right)}$$

Take the square root of 1960000.

$$x=\frac{-400±1400}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-400±1400}{-20}$$

Now solve the equation x=\frac{-400±1400}{-20} when ± is plus. Add -400 to 1400.

$$x=\frac{1000}{-20}$$

Divide 1000 by -20.

$$x=-50$$

Now solve the equation x=\frac{-400±1400}{-20} when ± is minus. Subtract 1400 from -400.

$$x=-\frac{1800}{-20}$$

Divide -1800 by -20.

$$x=90$$

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 90 for x_{2}.

$$-10x^{2}+400x+45000=-10\left(x-\left(-50\right)\right)\left(x-90\right)$$

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

$$-10x^{2}+400x+45000=-10\left(x+50\right)\left(x-90\right)$$

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.

$$x ^ 2 -40x -4500 = 0$$

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

$$r + s = 40 rs = -4500$$

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 = -4500

$$(20 - u) (20 + u) = -4500$$

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

$$400 - u^2 = -4500$$

Simplify the expression by subtracting 400 on both sides

$$-u^2 = -4500-400 = -4900$$

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

$$u^2 = 4900 u = \pm\sqrt{4900} = \pm 70$$

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

$$r =20 - 70 = -50 s = 20 + 70 = 90$$

Evaluate

$$-10\left(x-90\right)\left(x+50\right)$$