Solve 7000+250x-5x^2

$$7000 + 250 x - 5 x ^ { 2 }$$

Factor

$$-5\left(x-70\right)\left(x+20\right)$$

Solution Steps

Factor out 5.

$$5\left(1400+50x-x^{2}\right)$$

Consider 1400+50x-x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+50x+1400$$

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+1400. To find a and b, set up a system to be solved.

$$a+b=50 ab=-1400=-1400$$

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 -1400.

$$-1,1400 -2,700 -4,350 -5,280 -7,200 -8,175 -10,140 -14,100 -20,70 -25,56 -28,50 -35,40$$

Calculate the sum for each pair.

$$-1+1400=1399 -2+700=698 -4+350=346 -5+280=275 -7+200=193 -8+175=167 -10+140=130 -14+100=86 -20+70=50 -25+56=31 -28+50=22 -35+40=5$$

The solution is the pair that gives sum 50.

$$a=70 b=-20$$

Rewrite -x^{2}+50x+1400 as \left(-x^{2}+70x\right)+\left(-20x+1400\right).

$$\left(-x^{2}+70x\right)+\left(-20x+1400\right)$$

Factor out -x in the first and -20 in the second group.

$$-x\left(x-70\right)-20\left(x-70\right)$$

Factor out common term x-70 by using distributive property.

$$\left(x-70\right)\left(-x-20\right)$$

Rewrite the complete factored expression.

$$5\left(x-70\right)\left(-x-20\right)$$

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.

$$-5x^{2}+250x+7000=0$$

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.

$$x=\frac{-250±\sqrt{250^{2}-4\left(-5\right)\times 7000}}{2\left(-5\right)}$$

Square 250.

$$x=\frac{-250±\sqrt{62500-4\left(-5\right)\times 7000}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-250±\sqrt{62500+20\times 7000}}{2\left(-5\right)}$$

Multiply 20 times 7000.

$$x=\frac{-250±\sqrt{62500+140000}}{2\left(-5\right)}$$

Add 62500 to 140000.

$$x=\frac{-250±\sqrt{202500}}{2\left(-5\right)}$$

Take the square root of 202500.

$$x=\frac{-250±450}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-250±450}{-10}$$

Now solve the equation x=\frac{-250±450}{-10} when ± is plus. Add -250 to 450.

$$x=\frac{200}{-10}$$

Divide 200 by -10.

$$x=-20$$

Now solve the equation x=\frac{-250±450}{-10} when ± is minus. Subtract 450 from -250.

$$x=-\frac{700}{-10}$$

Divide -700 by -10.

$$x=70$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -20 for x_{1} and 70 for x_{2}.

$$-5x^{2}+250x+7000=-5\left(x-\left(-20\right)\right)\left(x-70\right)$$

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

$$-5x^{2}+250x+7000=-5\left(x+20\right)\left(x-70\right)$$

Evaluate

$$-5\left(x-70\right)\left(x+20\right)$$