Solve 4n^2-14n-15

$$4 n ^ { 2 } - 14 n - 15$$

Factor

$$4\left(n-\frac{7-\sqrt{109}}{4}\right)\left(n-\frac{\sqrt{109}+7}{4}\right)$$

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.

$$4n^{2}-14n-15=0$$

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.

$$n=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 4\left(-15\right)}}{2\times 4}$$

Square -14.

$$n=\frac{-\left(-14\right)±\sqrt{196-4\times 4\left(-15\right)}}{2\times 4}$$

Multiply -4 times 4.

$$n=\frac{-\left(-14\right)±\sqrt{196-16\left(-15\right)}}{2\times 4}$$

Multiply -16 times -15.

$$n=\frac{-\left(-14\right)±\sqrt{196+240}}{2\times 4}$$

Add 196 to 240.

$$n=\frac{-\left(-14\right)±\sqrt{436}}{2\times 4}$$

Take the square root of 436.

$$n=\frac{-\left(-14\right)±2\sqrt{109}}{2\times 4}$$

The opposite of -14 is 14.

$$n=\frac{14±2\sqrt{109}}{2\times 4}$$

Multiply 2 times 4.

$$n=\frac{14±2\sqrt{109}}{8}$$

Now solve the equation n=\frac{14±2\sqrt{109}}{8} when ± is plus. Add 14 to 2\sqrt{109}.

$$n=\frac{2\sqrt{109}+14}{8}$$

Divide 14+2\sqrt{109} by 8.

$$n=\frac{\sqrt{109}+7}{4}$$

Now solve the equation n=\frac{14±2\sqrt{109}}{8} when ± is minus. Subtract 2\sqrt{109} from 14.

$$n=\frac{14-2\sqrt{109}}{8}$$

Divide 14-2\sqrt{109} by 8.

$$n=\frac{7-\sqrt{109}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{109}}{4} for x_{1} and \frac{7-\sqrt{109}}{4} for x_{2}.

$$4n^{2}-14n-15=4\left(n-\frac{\sqrt{109}+7}{4}\right)\left(n-\frac{7-\sqrt{109}}{4}\right)$$

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 4

$$x ^ 2 -\frac{7}{2}x -\frac{15}{4} = 0$$

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

$$r + s = \frac{7}{2} rs = -\frac{15}{4}$$

Two numbers r and s sum up to \frac{7}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{2} = \frac{7}{4}. 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>

$$r = \frac{7}{4} - u s = \frac{7}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{15}{4}

$$(\frac{7}{4} - u) (\frac{7}{4} + u) = -\frac{15}{4}$$

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

$$\frac{49}{16} - u^2 = -\frac{15}{4}$$

Simplify the expression by subtracting \frac{49}{16} on both sides

$$-u^2 = -\frac{15}{4}-\frac{49}{16} = -\frac{109}{16}$$

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

$$u^2 = \frac{109}{16} u = \pm\sqrt{\frac{109}{16}} = \pm \frac{\sqrt{109}}{4}$$

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

$$r =\frac{7}{4} - \frac{\sqrt{109}}{4} = -0.860 s = \frac{7}{4} + \frac{\sqrt{109}}{4} = 4.360$$

Evaluate

$$4n^{2}-14n-15$$