Solve 7000x^2-4880x+792

$$7000 { x }^{ 2 } -4880x+792$$

Factor

$$8\left(25x-11\right)\left(35x-9\right)$$

Solution Steps

Factor out 8.

$$8\left(875x^{2}-610x+99\right)$$

Consider 875x^{2}-610x+99. Factor the expression by grouping. First, the expression needs to be rewritten as 875x^{2}+ax+bx+99. To find a and b, set up a system to be solved.

$$a+b=-610 ab=875\times 99=86625$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 86625.

$$-1,-86625 -3,-28875 -5,-17325 -7,-12375 -9,-9625 -11,-7875 -15,-5775 -21,-4125 -25,-3465 -33,-2625 -35,-2475 -45,-1925 -55,-1575 -63,-1375 -75,-1155 -77,-1125 -99,-875 -105,-825 -125,-693 -165,-525 -175,-495 -225,-385 -231,-375 -275,-315$$

Calculate the sum for each pair.

$$-1-86625=-86626 -3-28875=-28878 -5-17325=-17330 -7-12375=-12382 -9-9625=-9634 -11-7875=-7886 -15-5775=-5790 -21-4125=-4146 -25-3465=-3490 -33-2625=-2658 -35-2475=-2510 -45-1925=-1970 -55-1575=-1630 -63-1375=-1438 -75-1155=-1230 -77-1125=-1202 -99-875=-974 -105-825=-930 -125-693=-818 -165-525=-690 -175-495=-670 -225-385=-610 -231-375=-606 -275-315=-590$$

The solution is the pair that gives sum -610.

$$a=-385 b=-225$$

Rewrite 875x^{2}-610x+99 as \left(875x^{2}-385x\right)+\left(-225x+99\right).

$$\left(875x^{2}-385x\right)+\left(-225x+99\right)$$

Factor out 35x in the first and -9 in the second group.

$$35x\left(25x-11\right)-9\left(25x-11\right)$$

Factor out common term 25x-11 by using distributive property.

$$\left(25x-11\right)\left(35x-9\right)$$

Rewrite the complete factored expression.

$$8\left(25x-11\right)\left(35x-9\right)$$

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.

$$7000x^{2}-4880x+792=0$$

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.

$$x=\frac{-\left(-4880\right)±\sqrt{\left(-4880\right)^{2}-4\times 7000\times 792}}{2\times 7000}$$

Square -4880.

$$x=\frac{-\left(-4880\right)±\sqrt{23814400-4\times 7000\times 792}}{2\times 7000}$$

Multiply -4 times 7000.

$$x=\frac{-\left(-4880\right)±\sqrt{23814400-28000\times 792}}{2\times 7000}$$

Multiply -28000 times 792.

$$x=\frac{-\left(-4880\right)±\sqrt{23814400-22176000}}{2\times 7000}$$

Add 23814400 to -22176000.

$$x=\frac{-\left(-4880\right)±\sqrt{1638400}}{2\times 7000}$$

Take the square root of 1638400.

$$x=\frac{-\left(-4880\right)±1280}{2\times 7000}$$

The opposite of -4880 is 4880.

$$x=\frac{4880±1280}{2\times 7000}$$

Multiply 2 times 7000.

$$x=\frac{4880±1280}{14000}$$

Now solve the equation x=\frac{4880±1280}{14000} when ± is plus. Add 4880 to 1280.

$$x=\frac{6160}{14000}$$

Reduce the fraction \frac{6160}{14000} to lowest terms by extracting and canceling out 560.

$$x=\frac{11}{25}$$

Now solve the equation x=\frac{4880±1280}{14000} when ± is minus. Subtract 1280 from 4880.

$$x=\frac{3600}{14000}$$

Reduce the fraction \frac{3600}{14000} to lowest terms by extracting and canceling out 400.

$$x=\frac{9}{35}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11}{25} for x_{1} and \frac{9}{35} for x_{2}.

$$7000x^{2}-4880x+792=7000\left(x-\frac{11}{25}\right)\left(x-\frac{9}{35}\right)$$

Subtract \frac{11}{25} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7000x^{2}-4880x+792=7000\times \frac{25x-11}{25}\left(x-\frac{9}{35}\right)$$

Subtract \frac{9}{35} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7000x^{2}-4880x+792=7000\times \frac{25x-11}{25}\times \frac{35x-9}{35}$$

Multiply \frac{25x-11}{25} times \frac{35x-9}{35} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$7000x^{2}-4880x+792=7000\times \frac{\left(25x-11\right)\left(35x-9\right)}{25\times 35}$$

Multiply 25 times 35.

$$7000x^{2}-4880x+792=7000\times \frac{\left(25x-11\right)\left(35x-9\right)}{875}$$

Cancel out 875, the greatest common factor in 7000 and 875.

$$7000x^{2}-4880x+792=8\left(25x-11\right)\left(35x-9\right)$$

Evaluate

$$7000x^{2}-4880x+792$$