Solve 10p^2-6p+1

$$10 p ^ { 2 } - 6 p + 1$$

Evaluate

$$10p^{2}-6p+1$$

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 10

$$x ^ 2 -\frac{3}{5}x +\frac{1}{10} = 0$$

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

$$r + s = \frac{3}{5} rs = \frac{1}{10}$$

Two numbers r and s sum up to \frac{3}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{3}{5} = \frac{3}{10}. 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>

$$r = \frac{3}{10} - u s = \frac{3}{10} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{10}

$$(\frac{3}{10} - u) (\frac{3}{10} + u) = \frac{1}{10}$$

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

$$\frac{9}{100} - u^2 = \frac{1}{10}$$

Simplify the expression by subtracting \frac{9}{100} on both sides

$$-u^2 = \frac{1}{10}-\frac{9}{100} = \frac{1}{100}$$

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

$$u^2 = -\frac{1}{100} u = \pm\sqrt{-\frac{1}{100}} = \pm \frac{1}{10}i$$

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

$$r =\frac{3}{10} - \frac{1}{10}i = 0.300 - 0.100i s = \frac{3}{10} + \frac{1}{10}i = 0.300 + 0.100i$$

Differentiate w.r.t. p

$$20p-6$$

Steps Using Derivative Rule for Sum

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2\times 10p^{2-1}-6p^{1-1}$$

Multiply 2 times 10.

$$20p^{2-1}-6p^{1-1}$$

Subtract 1 from 2.

$$20p^{1}-6p^{1-1}$$

Subtract 1 from 1.

$$20p^{1}-6p^{0}$$

For any term t, t^{1}=t.

$$20p-6p^{0}$$

For any term t except 0, t^{0}=1.

$$20p-6$$