Solve f(p)=p^2+7p+1

$$f ( p ) = p ^ { 2 } + 7 p + 1$$

Factor

$$\left(p-\frac{-3\sqrt{5}-7}{2}\right)\left(p-\frac{3\sqrt{5}-7}{2}\right)$$

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.

$$p^{2}+7p+1=0$$

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.

$$p=\frac{-7±\sqrt{7^{2}-4}}{2}$$

Square 7.

$$p=\frac{-7±\sqrt{49-4}}{2}$$

Add 49 to -4.

$$p=\frac{-7±\sqrt{45}}{2}$$

Take the square root of 45.

$$p=\frac{-7±3\sqrt{5}}{2}$$

Now solve the equation p=\frac{-7±3\sqrt{5}}{2} when ± is plus. Add -7 to 3\sqrt{5}.

$$p=\frac{3\sqrt{5}-7}{2}$$

Now solve the equation p=\frac{-7±3\sqrt{5}}{2} when ± is minus. Subtract 3\sqrt{5} from -7.

$$p=\frac{-3\sqrt{5}-7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+3\sqrt{5}}{2} for x_{1} and \frac{-7-3\sqrt{5}}{2} for x_{2}.

$$p^{2}+7p+1=\left(p-\frac{3\sqrt{5}-7}{2}\right)\left(p-\frac{-3\sqrt{5}-7}{2}\right)$$

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.

$$x ^ 2 +7x +1 = 0$$

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

$$r + s = -7 rs = 1$$

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

$$r = -\frac{7}{2} - u s = -\frac{7}{2} + u$$

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

$$(-\frac{7}{2} - u) (-\frac{7}{2} + u) = 1$$

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

$$\frac{49}{4} - u^2 = 1$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = 1-\frac{49}{4} = -\frac{45}{4}$$

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

$$u^2 = \frac{45}{4} u = \pm\sqrt{\frac{45}{4}} = \pm \frac{\sqrt{45}}{2}$$

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

$$r =-\frac{7}{2} - \frac{\sqrt{45}}{2} = -6.854 s = -\frac{7}{2} + \frac{\sqrt{45}}{2} = -0.146$$

Evaluate

$$p^{2}+7p+1$$