Solve -20x^2+330x-1000

$$-20 { x }^{ 2 } +330x-1000$$

Factor

$$10\left(4-x\right)\left(2x-25\right)$$

Solution Steps

Factor out 10.

$$10\left(-2x^{2}+33x-100\right)$$

Consider -2x^{2}+33x-100. Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx-100. To find a and b, set up a system to be solved.

$$a+b=33 ab=-2\left(-100\right)=200$$

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 200.

$$1,200 2,100 4,50 5,40 8,25 10,20$$

Calculate the sum for each pair.

$$1+200=201 2+100=102 4+50=54 5+40=45 8+25=33 10+20=30$$

The solution is the pair that gives sum 33.

$$a=25 b=8$$

Rewrite -2x^{2}+33x-100 as \left(-2x^{2}+25x\right)+\left(8x-100\right).

$$\left(-2x^{2}+25x\right)+\left(8x-100\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(2x-25\right)+4\left(2x-25\right)$$

Factor out common term 2x-25 by using distributive property.

$$\left(2x-25\right)\left(-x+4\right)$$

Rewrite the complete factored expression.

$$10\left(2x-25\right)\left(-x+4\right)$$

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.

$$-20x^{2}+330x-1000=0$$

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.

$$x=\frac{-330±\sqrt{330^{2}-4\left(-20\right)\left(-1000\right)}}{2\left(-20\right)}$$

Square 330.

$$x=\frac{-330±\sqrt{108900-4\left(-20\right)\left(-1000\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-330±\sqrt{108900+80\left(-1000\right)}}{2\left(-20\right)}$$

Multiply 80 times -1000.

$$x=\frac{-330±\sqrt{108900-80000}}{2\left(-20\right)}$$

Add 108900 to -80000.

$$x=\frac{-330±\sqrt{28900}}{2\left(-20\right)}$$

Take the square root of 28900.

$$x=\frac{-330±170}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-330±170}{-40}$$

Now solve the equation x=\frac{-330±170}{-40} when ± is plus. Add -330 to 170.

$$x=-\frac{160}{-40}$$

Divide -160 by -40.

$$x=4$$

Now solve the equation x=\frac{-330±170}{-40} when ± is minus. Subtract 170 from -330.

$$x=-\frac{500}{-40}$$

Reduce the fraction \frac{-500}{-40} to lowest terms by extracting and canceling out 20.

$$x=\frac{25}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and \frac{25}{2} for x_{2}.

$$-20x^{2}+330x-1000=-20\left(x-4\right)\left(x-\frac{25}{2}\right)$$

Subtract \frac{25}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-20x^{2}+330x-1000=-20\left(x-4\right)\times \frac{-2x+25}{-2}$$

Cancel out 2, the greatest common factor in -20 and 2.

$$-20x^{2}+330x-1000=10\left(x-4\right)\left(-2x+25\right)$$

Evaluate

$$-20x^{2}+330x-1000$$