Solve 24w^2-23w-630

$$24 w ^ { 2 } - 23 w - 630$$

Factor

$$\left(8w-45\right)\left(3w+14\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 24w^{2}+aw+bw-630. To find a and b, set up a system to be solved.

$$a+b=-23 ab=24\left(-630\right)=-15120$$

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 -15120.

$$1,-15120 2,-7560 3,-5040 4,-3780 5,-3024 6,-2520 7,-2160 8,-1890 9,-1680 10,-1512 12,-1260 14,-1080 15,-1008 16,-945 18,-840 20,-756 21,-720 24,-630 27,-560 28,-540 30,-504 35,-432 36,-420 40,-378 42,-360 45,-336 48,-315 54,-280 56,-270 60,-252 63,-240 70,-216 72,-210 80,-189 84,-180 90,-168 105,-144 108,-140 112,-135 120,-126$$

Calculate the sum for each pair.

$$1-15120=-15119 2-7560=-7558 3-5040=-5037 4-3780=-3776 5-3024=-3019 6-2520=-2514 7-2160=-2153 8-1890=-1882 9-1680=-1671 10-1512=-1502 12-1260=-1248 14-1080=-1066 15-1008=-993 16-945=-929 18-840=-822 20-756=-736 21-720=-699 24-630=-606 27-560=-533 28-540=-512 30-504=-474 35-432=-397 36-420=-384 40-378=-338 42-360=-318 45-336=-291 48-315=-267 54-280=-226 56-270=-214 60-252=-192 63-240=-177 70-216=-146 72-210=-138 80-189=-109 84-180=-96 90-168=-78 105-144=-39 108-140=-32 112-135=-23 120-126=-6$$

The solution is the pair that gives sum -23.

$$a=-135 b=112$$

Rewrite 24w^{2}-23w-630 as \left(24w^{2}-135w\right)+\left(112w-630\right).

$$\left(24w^{2}-135w\right)+\left(112w-630\right)$$

Factor out 3w in the first and 14 in the second group.

$$3w\left(8w-45\right)+14\left(8w-45\right)$$

Factor out common term 8w-45 by using distributive property.

$$\left(8w-45\right)\left(3w+14\right)$$

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.

$$24w^{2}-23w-630=0$$

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.

$$w=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 24\left(-630\right)}}{2\times 24}$$

Square -23.

$$w=\frac{-\left(-23\right)±\sqrt{529-4\times 24\left(-630\right)}}{2\times 24}$$

Multiply -4 times 24.

$$w=\frac{-\left(-23\right)±\sqrt{529-96\left(-630\right)}}{2\times 24}$$

Multiply -96 times -630.

$$w=\frac{-\left(-23\right)±\sqrt{529+60480}}{2\times 24}$$

Add 529 to 60480.

$$w=\frac{-\left(-23\right)±\sqrt{61009}}{2\times 24}$$

Take the square root of 61009.

$$w=\frac{-\left(-23\right)±247}{2\times 24}$$

The opposite of -23 is 23.

$$w=\frac{23±247}{2\times 24}$$

Multiply 2 times 24.

$$w=\frac{23±247}{48}$$

Now solve the equation w=\frac{23±247}{48} when ± is plus. Add 23 to 247.

$$w=\frac{270}{48}$$

Reduce the fraction \frac{270}{48} to lowest terms by extracting and canceling out 6.

$$w=\frac{45}{8}$$

Now solve the equation w=\frac{23±247}{48} when ± is minus. Subtract 247 from 23.

$$w=-\frac{224}{48}$$

Reduce the fraction \frac{-224}{48} to lowest terms by extracting and canceling out 16.

$$w=-\frac{14}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{45}{8} for x_{1} and -\frac{14}{3} for x_{2}.

$$24w^{2}-23w-630=24\left(w-\frac{45}{8}\right)\left(w-\left(-\frac{14}{3}\right)\right)$$

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

$$24w^{2}-23w-630=24\left(w-\frac{45}{8}\right)\left(w+\frac{14}{3}\right)$$

Subtract \frac{45}{8} from w by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$24w^{2}-23w-630=24\times \frac{8w-45}{8}\left(w+\frac{14}{3}\right)$$

Add \frac{14}{3} to w by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$24w^{2}-23w-630=24\times \frac{8w-45}{8}\times \frac{3w+14}{3}$$

Multiply \frac{8w-45}{8} times \frac{3w+14}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$24w^{2}-23w-630=24\times \frac{\left(8w-45\right)\left(3w+14\right)}{8\times 3}$$

Multiply 8 times 3.

$$24w^{2}-23w-630=24\times \frac{\left(8w-45\right)\left(3w+14\right)}{24}$$

Cancel out 24, the greatest common factor in 24 and 24.

$$24w^{2}-23w-630=\left(8w-45\right)\left(3w+14\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 24

$$x ^ 2 -\frac{23}{24}x -\frac{105}{4} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{23}{24} rs = -\frac{105}{4}$$

Two numbers r and s sum up to \frac{23}{24} exactly when the average of the two numbers is \frac{1}{2}*\frac{23}{24} = \frac{23}{48}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{23}{48} - u s = \frac{23}{48} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{105}{4}

$$(\frac{23}{48} - u) (\frac{23}{48} + u) = -\frac{105}{4}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{529}{2304} - u^2 = -\frac{105}{4}$$

Simplify the expression by subtracting \frac{529}{2304} on both sides

$$-u^2 = -\frac{105}{4}-\frac{529}{2304} = -\frac{61009}{2304}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{61009}{2304} u = \pm\sqrt{\frac{61009}{2304}} = \pm \frac{247}{48}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{23}{48} - \frac{247}{48} = -4.667 s = \frac{23}{48} + \frac{247}{48} = 5.625$$

Evaluate

$$\left(8w-45\right)\left(3w+14\right)$$