Solve 3x^2-32x-91=

$$3 x ^ { 2 } - 32 x - 91 =$$

Factor

$$\left(x-13\right)\left(3x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-91. To find a and b, set up a system to be solved.

$$a+b=-32 ab=3\left(-91\right)=-273$$

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 -273.

$$1,-273 3,-91 7,-39 13,-21$$

Calculate the sum for each pair.

$$1-273=-272 3-91=-88 7-39=-32 13-21=-8$$

The solution is the pair that gives sum -32.

$$a=-39 b=7$$

Rewrite 3x^{2}-32x-91 as \left(3x^{2}-39x\right)+\left(7x-91\right).

$$\left(3x^{2}-39x\right)+\left(7x-91\right)$$

Factor out 3x in the first and 7 in the second group.

$$3x\left(x-13\right)+7\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(3x+7\right)$$

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.

$$3x^{2}-32x-91=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 3\left(-91\right)}}{2\times 3}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 3\left(-91\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-32\right)±\sqrt{1024-12\left(-91\right)}}{2\times 3}$$

Multiply -12 times -91.

$$x=\frac{-\left(-32\right)±\sqrt{1024+1092}}{2\times 3}$$

Add 1024 to 1092.

$$x=\frac{-\left(-32\right)±\sqrt{2116}}{2\times 3}$$

Take the square root of 2116.

$$x=\frac{-\left(-32\right)±46}{2\times 3}$$

The opposite of -32 is 32.

$$x=\frac{32±46}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{32±46}{6}$$

Now solve the equation x=\frac{32±46}{6} when ± is plus. Add 32 to 46.

$$x=\frac{78}{6}$$

Divide 78 by 6.

$$x=13$$

Now solve the equation x=\frac{32±46}{6} when ± is minus. Subtract 46 from 32.

$$x=-\frac{14}{6}$$

Reduce the fraction \frac{-14}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{7}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13 for x_{1} and -\frac{7}{3} for x_{2}.

$$3x^{2}-32x-91=3\left(x-13\right)\left(x-\left(-\frac{7}{3}\right)\right)$$

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

$$3x^{2}-32x-91=3\left(x-13\right)\left(x+\frac{7}{3}\right)$$

Add \frac{7}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}-32x-91=3\left(x-13\right)\times \frac{3x+7}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3x^{2}-32x-91=\left(x-13\right)\left(3x+7\right)$$

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.This is achieved by dividing both sides of the equation by 3

$$x ^ 2 -\frac{32}{3}x -\frac{91}{3} = 0$$

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

$$r + s = \frac{32}{3} rs = -\frac{91}{3}$$

Two numbers r and s sum up to \frac{32}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{32}{3} = \frac{16}{3}. 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>

$$r = \frac{16}{3} - u s = \frac{16}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{91}{3}

$$(\frac{16}{3} - u) (\frac{16}{3} + u) = -\frac{91}{3}$$

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

$$\frac{256}{9} - u^2 = -\frac{91}{3}$$

Simplify the expression by subtracting \frac{256}{9} on both sides

$$-u^2 = -\frac{91}{3}-\frac{256}{9} = -\frac{529}{9}$$

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

$$u^2 = \frac{529}{9} u = \pm\sqrt{\frac{529}{9}} = \pm \frac{23}{3}$$

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

$$r =\frac{16}{3} - \frac{23}{3} = -2.333 s = \frac{16}{3} + \frac{23}{3} = 13$$

Evaluate

$$\left(x-13\right)\left(3x+7\right)$$