Solve 16y^2+15y-16

$$16 y ^ { 2 } + 15 y - 16$$

Factor

$$16\left(y-\frac{-\sqrt{1249}-15}{32}\right)\left(y-\frac{\sqrt{1249}-15}{32}\right)$$

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.

$$16y^{2}+15y-16=0$$

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.

$$y=\frac{-15±\sqrt{15^{2}-4\times 16\left(-16\right)}}{2\times 16}$$

Square 15.

$$y=\frac{-15±\sqrt{225-4\times 16\left(-16\right)}}{2\times 16}$$

Multiply -4 times 16.

$$y=\frac{-15±\sqrt{225-64\left(-16\right)}}{2\times 16}$$

Multiply -64 times -16.

$$y=\frac{-15±\sqrt{225+1024}}{2\times 16}$$

Add 225 to 1024.

$$y=\frac{-15±\sqrt{1249}}{2\times 16}$$

Multiply 2 times 16.

$$y=\frac{-15±\sqrt{1249}}{32}$$

Now solve the equation y=\frac{-15±\sqrt{1249}}{32} when ± is plus. Add -15 to \sqrt{1249}.

$$y=\frac{\sqrt{1249}-15}{32}$$

Now solve the equation y=\frac{-15±\sqrt{1249}}{32} when ± is minus. Subtract \sqrt{1249} from -15.

$$y=\frac{-\sqrt{1249}-15}{32}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-15+\sqrt{1249}}{32} for x_{1} and \frac{-15-\sqrt{1249}}{32} for x_{2}.

$$16y^{2}+15y-16=16\left(y-\frac{\sqrt{1249}-15}{32}\right)\left(y-\frac{-\sqrt{1249}-15}{32}\right)$$

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 16

$$x ^ 2 +\frac{15}{16}x -1 = 0$$

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

$$r + s = -\frac{15}{16} rs = -1$$

Two numbers r and s sum up to -\frac{15}{16} exactly when the average of the two numbers is \frac{1}{2}*-\frac{15}{16} = -\frac{15}{32}. 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>

$$r = -\frac{15}{32} - u s = -\frac{15}{32} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(-\frac{15}{32} - u) (-\frac{15}{32} + u) = -1$$

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

$$\frac{225}{1024} - u^2 = -1$$

Simplify the expression by subtracting \frac{225}{1024} on both sides

$$-u^2 = -1-\frac{225}{1024} = -\frac{1249}{1024}$$

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

$$u^2 = \frac{1249}{1024} u = \pm\sqrt{\frac{1249}{1024}} = \pm \frac{\sqrt{1249}}{32}$$

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

$$r =-\frac{15}{32} - \frac{\sqrt{1249}}{32} = -1.573 s = -\frac{15}{32} + \frac{\sqrt{1249}}{32} = 0.636$$

Evaluate

$$16y^{2}+15y-16$$