Solve 28x^2-87x-1485=0

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

Solve for x

$$x = -\frac{165}{28} = -5\frac{25}{28} \approx -5.892857143$$

$$x=9$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side 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.

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

To find equation solutions, solve x-9=0 and 28x+165=0.

$$x=9 x=-\frac{165}{28}$$

Steps Using the Quadratic Formula

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 28 for a, -87 for b, and -1485 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

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.

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

The equation is now solved.

$$x=9 x=-\frac{165}{28}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Add 1485 to both sides of the equation.

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

Subtracting -1485 from itself leaves 0.

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

Subtract -1485 from 0.

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

Divide both sides by 28.

$$\frac{28x^{2}-87x}{28}=\frac{1485}{28}$$

Dividing by 28 undoes the multiplication by 28.

$$x^{2}-\frac{87}{28}x=\frac{1485}{28}$$

Divide -\frac{87}{28}, the coefficient of the x term, by 2 to get -\frac{87}{56}. Then add the square of -\frac{87}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{87}{28}x+\left(-\frac{87}{56}\right)^{2}=\frac{1485}{28}+\left(-\frac{87}{56}\right)^{2}$$

Square -\frac{87}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{87}{28}x+\frac{7569}{3136}=\frac{1485}{28}+\frac{7569}{3136}$$

Add \frac{1485}{28} to \frac{7569}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{87}{28}x+\frac{7569}{3136}=\frac{173889}{3136}$$

Factor x^{2}-\frac{87}{28}x+\frac{7569}{3136}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{87}{56}\right)^{2}=\frac{173889}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{87}{56}\right)^{2}}=\sqrt{\frac{173889}{3136}}$$

Simplify.

$$x-\frac{87}{56}=\frac{417}{56} x-\frac{87}{56}=-\frac{417}{56}$$

Add \frac{87}{56} to both sides of the equation.

$$x=9 x=-\frac{165}{28}$$