Solve 28x^2-87x-1485

$$28 { x }^{ 2 } -87x-1485$$

Factor

$$\left(x-9\right)\left(28x+165\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 28x^{2}+ax+bx-1485. To find a and b, set up a system to be solved.

$$a+b=-87 ab=28\left(-1485\right)=-41580$$

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 -41580.

$$1,-41580 2,-20790 3,-13860 4,-10395 5,-8316 6,-6930 7,-5940 9,-4620 10,-4158 11,-3780 12,-3465 14,-2970 15,-2772 18,-2310 20,-2079 21,-1980 22,-1890 27,-1540 28,-1485 30,-1386 33,-1260 35,-1188 36,-1155 42,-990 44,-945 45,-924 54,-770 55,-756 60,-693 63,-660 66,-630 70,-594 77,-540 84,-495 90,-462 99,-420 105,-396 108,-385 110,-378 126,-330 132,-315 135,-308 140,-297 154,-270 165,-252 180,-231 189,-220 198,-210$$

Calculate the sum for each pair.

$$1-41580=-41579 2-20790=-20788 3-13860=-13857 4-10395=-10391 5-8316=-8311 6-6930=-6924 7-5940=-5933 9-4620=-4611 10-4158=-4148 11-3780=-3769 12-3465=-3453 14-2970=-2956 15-2772=-2757 18-2310=-2292 20-2079=-2059 21-1980=-1959 22-1890=-1868 27-1540=-1513 28-1485=-1457 30-1386=-1356 33-1260=-1227 35-1188=-1153 36-1155=-1119 42-990=-948 44-945=-901 45-924=-879 54-770=-716 55-756=-701 60-693=-633 63-660=-597 66-630=-564 70-594=-524 77-540=-463 84-495=-411 90-462=-372 99-420=-321 105-396=-291 108-385=-277 110-378=-268 126-330=-204 132-315=-183 135-308=-173 140-297=-157 154-270=-116 165-252=-87 180-231=-51 189-220=-31 198-210=-12$$

The solution is the pair that gives sum -87.

$$a=-252 b=165$$

Rewrite 28x^{2}-87x-1485 as \left(28x^{2}-252x\right)+\left(165x-1485\right).

$$\left(28x^{2}-252x\right)+\left(165x-1485\right)$$

Factor out 28x in the first and 165 in the second group.

$$28x\left(x-9\right)+165\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(28x+165\right)$$

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.

$$28x^{2}-87x-1485=0$$

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.

$$x=\frac{-\left(-87\right)±\sqrt{\left(-87\right)^{2}-4\times 28\left(-1485\right)}}{2\times 28}$$

Square -87.

$$x=\frac{-\left(-87\right)±\sqrt{7569-4\times 28\left(-1485\right)}}{2\times 28}$$

Multiply -4 times 28.

$$x=\frac{-\left(-87\right)±\sqrt{7569-112\left(-1485\right)}}{2\times 28}$$

Multiply -112 times -1485.

$$x=\frac{-\left(-87\right)±\sqrt{7569+166320}}{2\times 28}$$

Add 7569 to 166320.

$$x=\frac{-\left(-87\right)±\sqrt{173889}}{2\times 28}$$

Take the square root of 173889.

$$x=\frac{-\left(-87\right)±417}{2\times 28}$$

The opposite of -87 is 87.

$$x=\frac{87±417}{2\times 28}$$

Multiply 2 times 28.

$$x=\frac{87±417}{56}$$

Now solve the equation x=\frac{87±417}{56} when ± is plus. Add 87 to 417.

$$x=\frac{504}{56}$$

Divide 504 by 56.

$$x=9$$

Now solve the equation x=\frac{87±417}{56} when ± is minus. Subtract 417 from 87.

$$x=-\frac{330}{56}$$

Reduce the fraction \frac{-330}{56} to lowest terms by extracting and canceling out 2.

$$x=-\frac{165}{28}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and -\frac{165}{28} for x_{2}.

$$28x^{2}-87x-1485=28\left(x-9\right)\left(x-\left(-\frac{165}{28}\right)\right)$$

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

$$28x^{2}-87x-1485=28\left(x-9\right)\left(x+\frac{165}{28}\right)$$

Add \frac{165}{28} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$28x^{2}-87x-1485=28\left(x-9\right)\times \frac{28x+165}{28}$$

Cancel out 28, the greatest common factor in 28 and 28.

$$28x^{2}-87x-1485=\left(x-9\right)\left(28x+165\right)$$

Evaluate

$$\left(x-9\right)\left(28x+165\right)$$