Solve x^2+24x-5040=

$${ x }^{ 2 } +24x-5040=$$

Factor

$$\left(x-60\right)\left(x+84\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-5040. To find a and b, set up a system to be solved.

$$a+b=24 ab=1\left(-5040\right)=-5040$$

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 -5040.

$$-1,5040 -2,2520 -3,1680 -4,1260 -5,1008 -6,840 -7,720 -8,630 -9,560 -10,504 -12,420 -14,360 -15,336 -16,315 -18,280 -20,252 -21,240 -24,210 -28,180 -30,168 -35,144 -36,140 -40,126 -42,120 -45,112 -48,105 -56,90 -60,84 -63,80 -70,72$$

Calculate the sum for each pair.

$$-1+5040=5039 -2+2520=2518 -3+1680=1677 -4+1260=1256 -5+1008=1003 -6+840=834 -7+720=713 -8+630=622 -9+560=551 -10+504=494 -12+420=408 -14+360=346 -15+336=321 -16+315=299 -18+280=262 -20+252=232 -21+240=219 -24+210=186 -28+180=152 -30+168=138 -35+144=109 -36+140=104 -40+126=86 -42+120=78 -45+112=67 -48+105=57 -56+90=34 -60+84=24 -63+80=17 -70+72=2$$

The solution is the pair that gives sum 24.

$$a=-60 b=84$$

Rewrite x^{2}+24x-5040 as \left(x^{2}-60x\right)+\left(84x-5040\right).

$$\left(x^{2}-60x\right)+\left(84x-5040\right)$$

Factor out x in the first and 84 in the second group.

$$x\left(x-60\right)+84\left(x-60\right)$$

Factor out common term x-60 by using distributive property.

$$\left(x-60\right)\left(x+84\right)$$

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.

$$x^{2}+24x-5040=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-5040\right)}}{2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-5040\right)}}{2}$$

Multiply -4 times -5040.

$$x=\frac{-24±\sqrt{576+20160}}{2}$$

Add 576 to 20160.

$$x=\frac{-24±\sqrt{20736}}{2}$$

Take the square root of 20736.

$$x=\frac{-24±144}{2}$$

Now solve the equation x=\frac{-24±144}{2} when ± is plus. Add -24 to 144.

$$x=\frac{120}{2}$$

Divide 120 by 2.

$$x=60$$

Now solve the equation x=\frac{-24±144}{2} when ± is minus. Subtract 144 from -24.

$$x=-\frac{168}{2}$$

Divide -168 by 2.

$$x=-84$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 60 for x_{1} and -84 for x_{2}.

$$x^{2}+24x-5040=\left(x-60\right)\left(x-\left(-84\right)\right)$$

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

$$x^{2}+24x-5040=\left(x-60\right)\left(x+84\right)$$

Evaluate

$$\left(x-60\right)\left(x+84\right)$$