Solve f(x)=-16x^2+14x+10

$$f ( x ) = - 16 x ^ { 2 } + 14 x + 10$$

Factor

$$-16\left(x-\frac{7-\sqrt{209}}{16}\right)\left(x-\frac{\sqrt{209}+7}{16}\right)$$

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.

$$-16x^{2}+14x+10=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-16\right)\times 10}}{2\left(-16\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-16\right)\times 10}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-14±\sqrt{196+64\times 10}}{2\left(-16\right)}$$

Multiply 64 times 10.

$$x=\frac{-14±\sqrt{196+640}}{2\left(-16\right)}$$

Add 196 to 640.

$$x=\frac{-14±\sqrt{836}}{2\left(-16\right)}$$

Take the square root of 836.

$$x=\frac{-14±2\sqrt{209}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-14±2\sqrt{209}}{-32}$$

Now solve the equation x=\frac{-14±2\sqrt{209}}{-32} when ± is plus. Add -14 to 2\sqrt{209}.

$$x=\frac{2\sqrt{209}-14}{-32}$$

Divide -14+2\sqrt{209} by -32.

$$x=\frac{7-\sqrt{209}}{16}$$

Now solve the equation x=\frac{-14±2\sqrt{209}}{-32} when ± is minus. Subtract 2\sqrt{209} from -14.

$$x=\frac{-2\sqrt{209}-14}{-32}$$

Divide -14-2\sqrt{209} by -32.

$$x=\frac{\sqrt{209}+7}{16}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7-\sqrt{209}}{16} for x_{1} and \frac{7+\sqrt{209}}{16} for x_{2}.

$$-16x^{2}+14x+10=-16\left(x-\frac{7-\sqrt{209}}{16}\right)\left(x-\frac{\sqrt{209}+7}{16}\right)$$

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.

$$x ^ 2 -\frac{7}{8}x -\frac{5}{8} = 0$$

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

$$r + s = \frac{7}{8} rs = -\frac{5}{8}$$

Two numbers r and s sum up to \frac{7}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{8} = \frac{7}{16}. 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>

$$r = \frac{7}{16} - u s = \frac{7}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{8}

$$(\frac{7}{16} - u) (\frac{7}{16} + u) = -\frac{5}{8}$$

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

$$\frac{49}{256} - u^2 = -\frac{5}{8}$$

Simplify the expression by subtracting \frac{49}{256} on both sides

$$-u^2 = -\frac{5}{8}-\frac{49}{256} = -\frac{209}{256}$$

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

$$u^2 = \frac{209}{256} u = \pm\sqrt{\frac{209}{256}} = \pm \frac{\sqrt{209}}{16}$$

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

$$r =\frac{7}{16} - \frac{\sqrt{209}}{16} = -0.466 s = \frac{7}{16} + \frac{\sqrt{209}}{16} = 1.341$$

Evaluate

$$10+14x-16x^{2}$$