Solve 11x^2-2x-48

$$11 x ^ { 2 } - 2 x - 48$$

Factor

$$\left(11x-24\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 11x^{2}+ax+bx-48. To find a and b, set up a system to be solved.

$$a+b=-2 ab=11\left(-48\right)=-528$$

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 -528.

$$1,-528 2,-264 3,-176 4,-132 6,-88 8,-66 11,-48 12,-44 16,-33 22,-24$$

Calculate the sum for each pair.

$$1-528=-527 2-264=-262 3-176=-173 4-132=-128 6-88=-82 8-66=-58 11-48=-37 12-44=-32 16-33=-17 22-24=-2$$

The solution is the pair that gives sum -2.

$$a=-24 b=22$$

Rewrite 11x^{2}-2x-48 as \left(11x^{2}-24x\right)+\left(22x-48\right).

$$\left(11x^{2}-24x\right)+\left(22x-48\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(11x-24\right)+2\left(11x-24\right)$$

Factor out common term 11x-24 by using distributive property.

$$\left(11x-24\right)\left(x+2\right)$$

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.

$$11x^{2}-2x-48=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 11\left(-48\right)}}{2\times 11}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 11\left(-48\right)}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-2\right)±\sqrt{4-44\left(-48\right)}}{2\times 11}$$

Multiply -44 times -48.

$$x=\frac{-\left(-2\right)±\sqrt{4+2112}}{2\times 11}$$

Add 4 to 2112.

$$x=\frac{-\left(-2\right)±\sqrt{2116}}{2\times 11}$$

Take the square root of 2116.

$$x=\frac{-\left(-2\right)±46}{2\times 11}$$

The opposite of -2 is 2.

$$x=\frac{2±46}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{2±46}{22}$$

Now solve the equation x=\frac{2±46}{22} when ± is plus. Add 2 to 46.

$$x=\frac{48}{22}$$

Reduce the fraction \frac{48}{22} to lowest terms by extracting and canceling out 2.

$$x=\frac{24}{11}$$

Now solve the equation x=\frac{2±46}{22} when ± is minus. Subtract 46 from 2.

$$x=-\frac{44}{22}$$

Divide -44 by 22.

$$x=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{24}{11} for x_{1} and -2 for x_{2}.

$$11x^{2}-2x-48=11\left(x-\frac{24}{11}\right)\left(x-\left(-2\right)\right)$$

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

$$11x^{2}-2x-48=11\left(x-\frac{24}{11}\right)\left(x+2\right)$$

Subtract \frac{24}{11} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$11x^{2}-2x-48=11\times \frac{11x-24}{11}\left(x+2\right)$$

Cancel out 11, the greatest common factor in 11 and 11.

$$11x^{2}-2x-48=\left(11x-24\right)\left(x+2\right)$$

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 11

$$x ^ 2 -\frac{2}{11}x -\frac{48}{11} = 0$$

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

$$r + s = \frac{2}{11} rs = -\frac{48}{11}$$

Two numbers r and s sum up to \frac{2}{11} exactly when the average of the two numbers is \frac{1}{2}*\frac{2}{11} = \frac{1}{11}. 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>

$$r = \frac{1}{11} - u s = \frac{1}{11} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{48}{11}

$$(\frac{1}{11} - u) (\frac{1}{11} + u) = -\frac{48}{11}$$

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

$$\frac{1}{121} - u^2 = -\frac{48}{11}$$

Simplify the expression by subtracting \frac{1}{121} on both sides

$$-u^2 = -\frac{48}{11}-\frac{1}{121} = -\frac{529}{121}$$

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

$$u^2 = \frac{529}{121} u = \pm\sqrt{\frac{529}{121}} = \pm \frac{23}{11}$$

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

$$r =\frac{1}{11} - \frac{23}{11} = -2 s = \frac{1}{11} + \frac{23}{11} = 2.182$$

Evaluate

$$\left(11x-24\right)\left(x+2\right)$$