Solve 15m^2+11m-19

$$15 m ^ { 2 } + 11 m - 19$$

Factor

$$15\left(m-\frac{-\sqrt{1261}-11}{30}\right)\left(m-\frac{\sqrt{1261}-11}{30}\right)$$

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.

$$15m^{2}+11m-19=0$$

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.

$$m=\frac{-11±\sqrt{11^{2}-4\times 15\left(-19\right)}}{2\times 15}$$

Square 11.

$$m=\frac{-11±\sqrt{121-4\times 15\left(-19\right)}}{2\times 15}$$

Multiply -4 times 15.

$$m=\frac{-11±\sqrt{121-60\left(-19\right)}}{2\times 15}$$

Multiply -60 times -19.

$$m=\frac{-11±\sqrt{121+1140}}{2\times 15}$$

Add 121 to 1140.

$$m=\frac{-11±\sqrt{1261}}{2\times 15}$$

Multiply 2 times 15.

$$m=\frac{-11±\sqrt{1261}}{30}$$

Now solve the equation m=\frac{-11±\sqrt{1261}}{30} when ± is plus. Add -11 to \sqrt{1261}.

$$m=\frac{\sqrt{1261}-11}{30}$$

Now solve the equation m=\frac{-11±\sqrt{1261}}{30} when ± is minus. Subtract \sqrt{1261} from -11.

$$m=\frac{-\sqrt{1261}-11}{30}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-11+\sqrt{1261}}{30} for x_{1} and \frac{-11-\sqrt{1261}}{30} for x_{2}.

$$15m^{2}+11m-19=15\left(m-\frac{\sqrt{1261}-11}{30}\right)\left(m-\frac{-\sqrt{1261}-11}{30}\right)$$

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 15

$$x ^ 2 +\frac{11}{15}x -\frac{19}{15} = 0$$

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

$$r + s = -\frac{11}{15} rs = -\frac{19}{15}$$

Two numbers r and s sum up to -\frac{11}{15} exactly when the average of the two numbers is \frac{1}{2}*-\frac{11}{15} = -\frac{11}{30}. 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>

$$r = -\frac{11}{30} - u s = -\frac{11}{30} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{19}{15}

$$(-\frac{11}{30} - u) (-\frac{11}{30} + u) = -\frac{19}{15}$$

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

$$\frac{121}{900} - u^2 = -\frac{19}{15}$$

Simplify the expression by subtracting \frac{121}{900} on both sides

$$-u^2 = -\frac{19}{15}-\frac{121}{900} = -\frac{1261}{900}$$

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

$$u^2 = \frac{1261}{900} u = \pm\sqrt{\frac{1261}{900}} = \pm \frac{\sqrt{1261}}{30}$$

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

$$r =-\frac{11}{30} - \frac{\sqrt{1261}}{30} = -1.550 s = -\frac{11}{30} + \frac{\sqrt{1261}}{30} = 0.817$$

Evaluate

$$15m^{2}+11m-19$$