Solve 13x^2+20x-92

$$13 x ^ { 2 } + 20 x - 92$$

Factor

$$\left(x-2\right)\left(13x+46\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 13x^{2}+ax+bx-92. To find a and b, set up a system to be solved.

$$a+b=20 ab=13\left(-92\right)=-1196$$

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 -1196.

$$-1,1196 -2,598 -4,299 -13,92 -23,52 -26,46$$

Calculate the sum for each pair.

$$-1+1196=1195 -2+598=596 -4+299=295 -13+92=79 -23+52=29 -26+46=20$$

The solution is the pair that gives sum 20.

$$a=-26 b=46$$

Rewrite 13x^{2}+20x-92 as \left(13x^{2}-26x\right)+\left(46x-92\right).

$$\left(13x^{2}-26x\right)+\left(46x-92\right)$$

Factor out 13x in the first and 46 in the second group.

$$13x\left(x-2\right)+46\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(13x+46\right)$$

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.

$$13x^{2}+20x-92=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\times 13\left(-92\right)}}{2\times 13}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 13\left(-92\right)}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-20±\sqrt{400-52\left(-92\right)}}{2\times 13}$$

Multiply -52 times -92.

$$x=\frac{-20±\sqrt{400+4784}}{2\times 13}$$

Add 400 to 4784.

$$x=\frac{-20±\sqrt{5184}}{2\times 13}$$

Take the square root of 5184.

$$x=\frac{-20±72}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{-20±72}{26}$$

Now solve the equation x=\frac{-20±72}{26} when ± is plus. Add -20 to 72.

$$x=\frac{52}{26}$$

Divide 52 by 26.

$$x=2$$

Now solve the equation x=\frac{-20±72}{26} when ± is minus. Subtract 72 from -20.

$$x=-\frac{92}{26}$$

Reduce the fraction \frac{-92}{26} to lowest terms by extracting and canceling out 2.

$$x=-\frac{46}{13}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -\frac{46}{13} for x_{2}.

$$13x^{2}+20x-92=13\left(x-2\right)\left(x-\left(-\frac{46}{13}\right)\right)$$

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

$$13x^{2}+20x-92=13\left(x-2\right)\left(x+\frac{46}{13}\right)$$

Add \frac{46}{13} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$13x^{2}+20x-92=13\left(x-2\right)\times \frac{13x+46}{13}$$

Cancel out 13, the greatest common factor in 13 and 13.

$$13x^{2}+20x-92=\left(x-2\right)\left(13x+46\right)$$

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 13

$$x ^ 2 +\frac{20}{13}x -\frac{92}{13} = 0$$

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

$$r + s = -\frac{20}{13} rs = -\frac{92}{13}$$

Two numbers r and s sum up to -\frac{20}{13} exactly when the average of the two numbers is \frac{1}{2}*-\frac{20}{13} = -\frac{10}{13}. 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>

$$r = -\frac{10}{13} - u s = -\frac{10}{13} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{92}{13}

$$(-\frac{10}{13} - u) (-\frac{10}{13} + u) = -\frac{92}{13}$$

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

$$\frac{100}{169} - u^2 = -\frac{92}{13}$$

Simplify the expression by subtracting \frac{100}{169} on both sides

$$-u^2 = -\frac{92}{13}-\frac{100}{169} = -\frac{1296}{169}$$

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

$$u^2 = \frac{1296}{169} u = \pm\sqrt{\frac{1296}{169}} = \pm \frac{36}{13}$$

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

$$r =-\frac{10}{13} - \frac{36}{13} = -3.538 s = -\frac{10}{13} + \frac{36}{13} = 2$$

Evaluate

$$\left(x-2\right)\left(13x+46\right)$$