Solve M(x)=-25x^2+100x+8000

$$M ( x ) = - 25 x ^ { 2 } + 100 x + 8000$$

Factor

$$-25\left(x-20\right)\left(x+16\right)$$

Solution Steps

Factor out 25.

$$25\left(-x^{2}+4x+320\right)$$

Consider -x^{2}+4x+320. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+320. To find a and b, set up a system to be solved.

$$a+b=4 ab=-320=-320$$

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 -320.

$$-1,320 -2,160 -4,80 -5,64 -8,40 -10,32 -16,20$$

Calculate the sum for each pair.

$$-1+320=319 -2+160=158 -4+80=76 -5+64=59 -8+40=32 -10+32=22 -16+20=4$$

The solution is the pair that gives sum 4.

$$a=20 b=-16$$

Rewrite -x^{2}+4x+320 as \left(-x^{2}+20x\right)+\left(-16x+320\right).

$$\left(-x^{2}+20x\right)+\left(-16x+320\right)$$

Factor out -x in the first and -16 in the second group.

$$-x\left(x-20\right)-16\left(x-20\right)$$

Factor out common term x-20 by using distributive property.

$$\left(x-20\right)\left(-x-16\right)$$

Rewrite the complete factored expression.

$$25\left(x-20\right)\left(-x-16\right)$$

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.

$$-25x^{2}+100x+8000=0$$

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.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-25\right)\times 8000}}{2\left(-25\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-25\right)\times 8000}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-100±\sqrt{10000+100\times 8000}}{2\left(-25\right)}$$

Multiply 100 times 8000.

$$x=\frac{-100±\sqrt{10000+800000}}{2\left(-25\right)}$$

Add 10000 to 800000.

$$x=\frac{-100±\sqrt{810000}}{2\left(-25\right)}$$

Take the square root of 810000.

$$x=\frac{-100±900}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{-100±900}{-50}$$

Now solve the equation x=\frac{-100±900}{-50} when ± is plus. Add -100 to 900.

$$x=\frac{800}{-50}$$

Divide 800 by -50.

$$x=-16$$

Now solve the equation x=\frac{-100±900}{-50} when ± is minus. Subtract 900 from -100.

$$x=-\frac{1000}{-50}$$

Divide -1000 by -50.

$$x=20$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -16 for x_{1} and 20 for x_{2}.

$$-25x^{2}+100x+8000=-25\left(x-\left(-16\right)\right)\left(x-20\right)$$

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

$$-25x^{2}+100x+8000=-25\left(x+16\right)\left(x-20\right)$$

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.

$$x ^ 2 -4x -320 = 0$$

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

$$r + s = 4 rs = -320$$

Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. 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>

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -320

$$(2 - u) (2 + u) = -320$$

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

$$4 - u^2 = -320$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -320-4 = -324$$

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

$$u^2 = 324 u = \pm\sqrt{324} = \pm 18$$

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

$$r =2 - 18 = -16 s = 2 + 18 = 20$$

Evaluate

$$-25\left(x-20\right)\left(x+16\right)$$