Solve 11m^2-31m-10

$$11 m ^ { 2 } - 31 m - 10$$

Factor

$$11\left(m-\frac{31-\sqrt{1401}}{22}\right)\left(m-\frac{\sqrt{1401}+31}{22}\right)$$

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.

$$11m^{2}-31m-10=0$$

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.

$$m=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 11\left(-10\right)}}{2\times 11}$$

Square -31.

$$m=\frac{-\left(-31\right)±\sqrt{961-4\times 11\left(-10\right)}}{2\times 11}$$

Multiply -4 times 11.

$$m=\frac{-\left(-31\right)±\sqrt{961-44\left(-10\right)}}{2\times 11}$$

Multiply -44 times -10.

$$m=\frac{-\left(-31\right)±\sqrt{961+440}}{2\times 11}$$

Add 961 to 440.

$$m=\frac{-\left(-31\right)±\sqrt{1401}}{2\times 11}$$

The opposite of -31 is 31.

$$m=\frac{31±\sqrt{1401}}{2\times 11}$$

Multiply 2 times 11.

$$m=\frac{31±\sqrt{1401}}{22}$$

Now solve the equation m=\frac{31±\sqrt{1401}}{22} when ± is plus. Add 31 to \sqrt{1401}.

$$m=\frac{\sqrt{1401}+31}{22}$$

Now solve the equation m=\frac{31±\sqrt{1401}}{22} when ± is minus. Subtract \sqrt{1401} from 31.

$$m=\frac{31-\sqrt{1401}}{22}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{31+\sqrt{1401}}{22} for x_{1} and \frac{31-\sqrt{1401}}{22} for x_{2}.

$$11m^{2}-31m-10=11\left(m-\frac{\sqrt{1401}+31}{22}\right)\left(m-\frac{31-\sqrt{1401}}{22}\right)$$

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

$$x ^ 2 -\frac{31}{11}x -\frac{10}{11} = 0$$

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

$$r + s = \frac{31}{11} rs = -\frac{10}{11}$$

Two numbers r and s sum up to \frac{31}{11} exactly when the average of the two numbers is \frac{1}{2}*\frac{31}{11} = \frac{31}{22}. 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>

$$r = \frac{31}{22} - u s = \frac{31}{22} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{10}{11}

$$(\frac{31}{22} - u) (\frac{31}{22} + u) = -\frac{10}{11}$$

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

$$\frac{961}{484} - u^2 = -\frac{10}{11}$$

Simplify the expression by subtracting \frac{961}{484} on both sides

$$-u^2 = -\frac{10}{11}-\frac{961}{484} = -\frac{1401}{484}$$

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

$$u^2 = \frac{1401}{484} u = \pm\sqrt{\frac{1401}{484}} = \pm \frac{\sqrt{1401}}{22}$$

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

$$r =\frac{31}{22} - \frac{\sqrt{1401}}{22} = -0.292 s = \frac{31}{22} + \frac{\sqrt{1401}}{22} = 3.110$$

Evaluate

$$11m^{2}-31m-10$$