Solve z(s)=-16s^2+15s+1451

$$z ( s ) = - 16 s ^ { 2 } + 15 s + 1451$$

Factor

$$-16\left(s-\frac{15-\sqrt{93089}}{32}\right)\left(s-\frac{\sqrt{93089}+15}{32}\right)$$

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.

$$-16s^{2}+15s+1451=0$$

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.

$$s=\frac{-15±\sqrt{15^{2}-4\left(-16\right)\times 1451}}{2\left(-16\right)}$$

Square 15.

$$s=\frac{-15±\sqrt{225-4\left(-16\right)\times 1451}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$s=\frac{-15±\sqrt{225+64\times 1451}}{2\left(-16\right)}$$

Multiply 64 times 1451.

$$s=\frac{-15±\sqrt{225+92864}}{2\left(-16\right)}$$

Add 225 to 92864.

$$s=\frac{-15±\sqrt{93089}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$s=\frac{-15±\sqrt{93089}}{-32}$$

Now solve the equation s=\frac{-15±\sqrt{93089}}{-32} when ± is plus. Add -15 to \sqrt{93089}.

$$s=\frac{\sqrt{93089}-15}{-32}$$

Divide -15+\sqrt{93089} by -32.

$$s=\frac{15-\sqrt{93089}}{32}$$

Now solve the equation s=\frac{-15±\sqrt{93089}}{-32} when ± is minus. Subtract \sqrt{93089} from -15.

$$s=\frac{-\sqrt{93089}-15}{-32}$$

Divide -15-\sqrt{93089} by -32.

$$s=\frac{\sqrt{93089}+15}{32}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{15-\sqrt{93089}}{32} for x_{1} and \frac{15+\sqrt{93089}}{32} for x_{2}.

$$-16s^{2}+15s+1451=-16\left(s-\frac{15-\sqrt{93089}}{32}\right)\left(s-\frac{\sqrt{93089}+15}{32}\right)$$

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.

$$x ^ 2 -\frac{15}{16}x -\frac{1451}{16} = 0$$

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

$$r + s = \frac{15}{16} rs = -\frac{1451}{16}$$

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 = -\frac{1451}{16}

$$(\frac{15}{32} - u) (\frac{15}{32} + u) = -\frac{1451}{16}$$

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

$$\frac{225}{1024} - u^2 = -\frac{1451}{16}$$

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

$$-u^2 = -\frac{1451}{16}-\frac{225}{1024} = -\frac{93089}{1024}$$

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

$$u^2 = \frac{93089}{1024} u = \pm\sqrt{\frac{93089}{1024}} = \pm \frac{\sqrt{93089}}{32}$$

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

$$r =\frac{15}{32} - \frac{\sqrt{93089}}{32} = -9.066 s = \frac{15}{32} + \frac{\sqrt{93089}}{32} = 10.003$$

Evaluate

$$1451+15s-16s^{2}$$