Solve f(x)=x^2+14x+42

$$f ( x ) = x ^ { 2 } + 14 x + 42$$

Factor

$$\left(x-\left(-\sqrt{7}-7\right)\right)\left(x-\left(\sqrt{7}-7\right)\right)$$

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.

$$x^{2}+14x+42=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\times 42}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 42}}{2}$$

Multiply -4 times 42.

$$x=\frac{-14±\sqrt{196-168}}{2}$$

Add 196 to -168.

$$x=\frac{-14±\sqrt{28}}{2}$$

Take the square root of 28.

$$x=\frac{-14±2\sqrt{7}}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{7}}{2} when ± is plus. Add -14 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-14}{2}$$

Divide -14+2\sqrt{7} by 2.

$$x=\sqrt{7}-7$$

Now solve the equation x=\frac{-14±2\sqrt{7}}{2} when ± is minus. Subtract 2\sqrt{7} from -14.

$$x=\frac{-2\sqrt{7}-14}{2}$$

Divide -14-2\sqrt{7} by 2.

$$x=-\sqrt{7}-7$$

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

$$x^{2}+14x+42=\left(x-\left(\sqrt{7}-7\right)\right)\left(x-\left(-\sqrt{7}-7\right)\right)$$

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.

$$x ^ 2 +14x +42 = 0$$

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

$$r + s = -14 rs = 42$$

Two numbers r and s sum up to -14 exactly when the average of the two numbers is \frac{1}{2}*-14 = -7. 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>

$$r = -7 - u s = -7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 42

$$(-7 - u) (-7 + u) = 42$$

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

$$49 - u^2 = 42$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = 42-49 = -7$$

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

$$u^2 = 7 u = \pm\sqrt{7} = \pm \sqrt{7}$$

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

$$r =-7 - \sqrt{7} = -9.646 s = -7 + \sqrt{7} = -4.354$$

Evaluate

$$x^{2}+14x+42$$