Solve -48w^2+24w+105

$$- 48 w ^ { 2 } + 24 w + 105$$

Factor

$$3\left(-4w-5\right)\left(4w-7\right)$$

Solution Steps

Factor out 3.

$$3\left(-16w^{2}+8w+35\right)$$

Consider -16w^{2}+8w+35. Factor the expression by grouping. First, the expression needs to be rewritten as -16w^{2}+aw+bw+35. To find a and b, set up a system to be solved.

$$a+b=8 ab=-16\times 35=-560$$

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 -560.

$$-1,560 -2,280 -4,140 -5,112 -7,80 -8,70 -10,56 -14,40 -16,35 -20,28$$

Calculate the sum for each pair.

$$-1+560=559 -2+280=278 -4+140=136 -5+112=107 -7+80=73 -8+70=62 -10+56=46 -14+40=26 -16+35=19 -20+28=8$$

The solution is the pair that gives sum 8.

$$a=28 b=-20$$

Rewrite -16w^{2}+8w+35 as \left(-16w^{2}+28w\right)+\left(-20w+35\right).

$$\left(-16w^{2}+28w\right)+\left(-20w+35\right)$$

Factor out -4w in the first and -5 in the second group.

$$-4w\left(4w-7\right)-5\left(4w-7\right)$$

Factor out common term 4w-7 by using distributive property.

$$\left(4w-7\right)\left(-4w-5\right)$$

Rewrite the complete factored expression.

$$3\left(4w-7\right)\left(-4w-5\right)$$

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.

$$-48w^{2}+24w+105=0$$

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.

$$w=\frac{-24±\sqrt{24^{2}-4\left(-48\right)\times 105}}{2\left(-48\right)}$$

Square 24.

$$w=\frac{-24±\sqrt{576-4\left(-48\right)\times 105}}{2\left(-48\right)}$$

Multiply -4 times -48.

$$w=\frac{-24±\sqrt{576+192\times 105}}{2\left(-48\right)}$$

Multiply 192 times 105.

$$w=\frac{-24±\sqrt{576+20160}}{2\left(-48\right)}$$

Add 576 to 20160.

$$w=\frac{-24±\sqrt{20736}}{2\left(-48\right)}$$

Take the square root of 20736.

$$w=\frac{-24±144}{2\left(-48\right)}$$

Multiply 2 times -48.

$$w=\frac{-24±144}{-96}$$

Now solve the equation w=\frac{-24±144}{-96} when ± is plus. Add -24 to 144.

$$w=\frac{120}{-96}$$

Reduce the fraction \frac{120}{-96} to lowest terms by extracting and canceling out 24.

$$w=-\frac{5}{4}$$

Now solve the equation w=\frac{-24±144}{-96} when ± is minus. Subtract 144 from -24.

$$w=-\frac{168}{-96}$$

Reduce the fraction \frac{-168}{-96} to lowest terms by extracting and canceling out 24.

$$w=\frac{7}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{4} for x_{1} and \frac{7}{4} for x_{2}.

$$-48w^{2}+24w+105=-48\left(w-\left(-\frac{5}{4}\right)\right)\left(w-\frac{7}{4}\right)$$

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

$$-48w^{2}+24w+105=-48\left(w+\frac{5}{4}\right)\left(w-\frac{7}{4}\right)$$

Add \frac{5}{4} to w by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-48w^{2}+24w+105=-48\times \frac{-4w-5}{-4}\left(w-\frac{7}{4}\right)$$

Subtract \frac{7}{4} from w by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-48w^{2}+24w+105=-48\times \frac{-4w-5}{-4}\times \frac{-4w+7}{-4}$$

Multiply \frac{-4w-5}{-4} times \frac{-4w+7}{-4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-48w^{2}+24w+105=-48\times \frac{\left(-4w-5\right)\left(-4w+7\right)}{-4\left(-4\right)}$$

Multiply -4 times -4.

$$-48w^{2}+24w+105=-48\times \frac{\left(-4w-5\right)\left(-4w+7\right)}{16}$$

Cancel out 16, the greatest common factor in -48 and 16.

$$-48w^{2}+24w+105=-3\left(-4w-5\right)\left(-4w+7\right)$$

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.

$$x ^ 2 -\frac{1}{2}x -\frac{35}{16} = 0$$

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

$$r + s = \frac{1}{2} rs = -\frac{35}{16}$$

Two numbers r and s sum up to \frac{1}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{1}{2} = \frac{1}{4}. 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>

$$r = \frac{1}{4} - u s = \frac{1}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{35}{16}

$$(\frac{1}{4} - u) (\frac{1}{4} + u) = -\frac{35}{16}$$

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

$$\frac{1}{16} - u^2 = -\frac{35}{16}$$

Simplify the expression by subtracting \frac{1}{16} on both sides

$$-u^2 = -\frac{35}{16}-\frac{1}{16} = -\frac{9}{4}$$

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

$$u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{2}$$

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

$$r =\frac{1}{4} - \frac{3}{2} = -1.250 s = \frac{1}{4} + \frac{3}{2} = 1.750$$

Evaluate

$$105+24w-48w^{2}$$