Solve -250x^2-1250x+21000

$$- 250 x ^ { 2 } - 1250 x + 21000$$

Factor

$$250\left(7-x\right)\left(x+12\right)$$

Solution Steps

Factor out 250.

$$250\left(-x^{2}-5x+84\right)$$

Consider -x^{2}-5x+84. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+84. To find a and b, set up a system to be solved.

$$a+b=-5 ab=-84=-84$$

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 -84.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Calculate the sum for each pair.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

The solution is the pair that gives sum -5.

$$a=7 b=-12$$

Rewrite -x^{2}-5x+84 as \left(-x^{2}+7x\right)+\left(-12x+84\right).

$$\left(-x^{2}+7x\right)+\left(-12x+84\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(-x+7\right)+12\left(-x+7\right)$$

Factor out common term -x+7 by using distributive property.

$$\left(-x+7\right)\left(x+12\right)$$

Rewrite the complete factored expression.

$$250\left(-x+7\right)\left(x+12\right)$$

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.

$$-250x^{2}-1250x+21000=0$$

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.

$$x=\frac{-\left(-1250\right)±\sqrt{\left(-1250\right)^{2}-4\left(-250\right)\times 21000}}{2\left(-250\right)}$$

Square -1250.

$$x=\frac{-\left(-1250\right)±\sqrt{1562500-4\left(-250\right)\times 21000}}{2\left(-250\right)}$$

Multiply -4 times -250.

$$x=\frac{-\left(-1250\right)±\sqrt{1562500+1000\times 21000}}{2\left(-250\right)}$$

Multiply 1000 times 21000.

$$x=\frac{-\left(-1250\right)±\sqrt{1562500+21000000}}{2\left(-250\right)}$$

Add 1562500 to 21000000.

$$x=\frac{-\left(-1250\right)±\sqrt{22562500}}{2\left(-250\right)}$$

Take the square root of 22562500.

$$x=\frac{-\left(-1250\right)±4750}{2\left(-250\right)}$$

The opposite of -1250 is 1250.

$$x=\frac{1250±4750}{2\left(-250\right)}$$

Multiply 2 times -250.

$$x=\frac{1250±4750}{-500}$$

Now solve the equation x=\frac{1250±4750}{-500} when ± is plus. Add 1250 to 4750.

$$x=\frac{6000}{-500}$$

Divide 6000 by -500.

$$x=-12$$

Now solve the equation x=\frac{1250±4750}{-500} when ± is minus. Subtract 4750 from 1250.

$$x=-\frac{3500}{-500}$$

Divide -3500 by -500.

$$x=7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -12 for x_{1} and 7 for x_{2}.

$$-250x^{2}-1250x+21000=-250\left(x-\left(-12\right)\right)\left(x-7\right)$$

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

$$-250x^{2}-1250x+21000=-250\left(x+12\right)\left(x-7\right)$$

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.

$$x ^ 2 +5x -84 = 0$$

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

$$r + s = -5 rs = -84$$

Two numbers r and s sum up to -5 exactly when the average of the two numbers is \frac{1}{2}*-5 = -\frac{5}{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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{5}{2} - u s = -\frac{5}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -84

$$(-\frac{5}{2} - u) (-\frac{5}{2} + u) = -84$$

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

$$\frac{25}{4} - u^2 = -84$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = -84-\frac{25}{4} = -\frac{361}{4}$$

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

$$u^2 = \frac{361}{4} u = \pm\sqrt{\frac{361}{4}} = \pm \frac{19}{2}$$

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

$$r =-\frac{5}{2} - \frac{19}{2} = -12 s = -\frac{5}{2} + \frac{19}{2} = 7$$

Evaluate

$$250\left(7-x\right)\left(x+12\right)$$