Solve 8x^2+134x-1584

$$8 x ^ { 2 } + 134 x - 1584$$

Factor

$$2\left(x-8\right)\left(4x+99\right)$$

Solution Steps

Factor out 2.

$$2\left(4x^{2}+67x-792\right)$$

Consider 4x^{2}+67x-792. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-792. To find a and b, set up a system to be solved.

$$a+b=67 ab=4\left(-792\right)=-3168$$

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 -3168.

$$-1,3168 -2,1584 -3,1056 -4,792 -6,528 -8,396 -9,352 -11,288 -12,264 -16,198 -18,176 -22,144 -24,132 -32,99 -33,96 -36,88 -44,72 -48,66$$

Calculate the sum for each pair.

$$-1+3168=3167 -2+1584=1582 -3+1056=1053 -4+792=788 -6+528=522 -8+396=388 -9+352=343 -11+288=277 -12+264=252 -16+198=182 -18+176=158 -22+144=122 -24+132=108 -32+99=67 -33+96=63 -36+88=52 -44+72=28 -48+66=18$$

The solution is the pair that gives sum 67.

$$a=-32 b=99$$

Rewrite 4x^{2}+67x-792 as \left(4x^{2}-32x\right)+\left(99x-792\right).

$$\left(4x^{2}-32x\right)+\left(99x-792\right)$$

Factor out 4x in the first and 99 in the second group.

$$4x\left(x-8\right)+99\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(4x+99\right)$$

Rewrite the complete factored expression.

$$2\left(x-8\right)\left(4x+99\right)$$

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.

$$8x^{2}+134x-1584=0$$

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.

$$x=\frac{-134±\sqrt{134^{2}-4\times 8\left(-1584\right)}}{2\times 8}$$

Square 134.

$$x=\frac{-134±\sqrt{17956-4\times 8\left(-1584\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-134±\sqrt{17956-32\left(-1584\right)}}{2\times 8}$$

Multiply -32 times -1584.

$$x=\frac{-134±\sqrt{17956+50688}}{2\times 8}$$

Add 17956 to 50688.

$$x=\frac{-134±\sqrt{68644}}{2\times 8}$$

Take the square root of 68644.

$$x=\frac{-134±262}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-134±262}{16}$$

Now solve the equation x=\frac{-134±262}{16} when ± is plus. Add -134 to 262.

$$x=\frac{128}{16}$$

Divide 128 by 16.

$$x=8$$

Now solve the equation x=\frac{-134±262}{16} when ± is minus. Subtract 262 from -134.

$$x=-\frac{396}{16}$$

Reduce the fraction \frac{-396}{16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{99}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -\frac{99}{4} for x_{2}.

$$8x^{2}+134x-1584=8\left(x-8\right)\left(x-\left(-\frac{99}{4}\right)\right)$$

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

$$8x^{2}+134x-1584=8\left(x-8\right)\left(x+\frac{99}{4}\right)$$

Add \frac{99}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}+134x-1584=8\left(x-8\right)\times \frac{4x+99}{4}$$

Cancel out 4, the greatest common factor in 8 and 4.

$$8x^{2}+134x-1584=2\left(x-8\right)\left(4x+99\right)$$

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 8

$$x ^ 2 +\frac{67}{4}x -198 = 0$$

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

$$r + s = -\frac{67}{4} rs = -198$$

Two numbers r and s sum up to -\frac{67}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{67}{4} = -\frac{67}{8}. 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>

$$r = -\frac{67}{8} - u s = -\frac{67}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -198

$$(-\frac{67}{8} - u) (-\frac{67}{8} + u) = -198$$

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

$$\frac{4489}{64} - u^2 = -198$$

Simplify the expression by subtracting \frac{4489}{64} on both sides

$$-u^2 = -198-\frac{4489}{64} = -\frac{17161}{64}$$

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

$$u^2 = \frac{17161}{64} u = \pm\sqrt{\frac{17161}{64}} = \pm \frac{131}{8}$$

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

$$r =-\frac{67}{8} - \frac{131}{8} = -24.750 s = -\frac{67}{8} + \frac{131}{8} = 8$$

Evaluate

$$2\left(x-8\right)\left(4x+99\right)$$