Solve 14x^2+16x+2

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

Factor

$$2\left(x+1\right)\left(7x+1\right)$$

Solution Steps

Factor out 2.

$$2\left(7x^{2}+8x+1\right)$$

Consider 7x^{2}+8x+1. Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=8 ab=7\times 1=7$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=7$$

Rewrite 7x^{2}+8x+1 as \left(7x^{2}+x\right)+\left(7x+1\right).

$$\left(7x^{2}+x\right)+\left(7x+1\right)$$

Factor out x in 7x^{2}+x.

$$x\left(7x+1\right)+7x+1$$

Factor out common term 7x+1 by using distributive property.

$$\left(7x+1\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$2\left(7x+1\right)\left(x+1\right)$$

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.

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

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.

$$x=\frac{-16±\sqrt{16^{2}-4\times 14\times 2}}{2\times 14}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 14\times 2}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-16±\sqrt{256-56\times 2}}{2\times 14}$$

Multiply -56 times 2.

$$x=\frac{-16±\sqrt{256-112}}{2\times 14}$$

Add 256 to -112.

$$x=\frac{-16±\sqrt{144}}{2\times 14}$$

Take the square root of 144.

$$x=\frac{-16±12}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{-16±12}{28}$$

Now solve the equation x=\frac{-16±12}{28} when ± is plus. Add -16 to 12.

$$x=-\frac{4}{28}$$

Reduce the fraction \frac{-4}{28} to lowest terms by extracting and canceling out 4.

$$x=-\frac{1}{7}$$

Now solve the equation x=\frac{-16±12}{28} when ± is minus. Subtract 12 from -16.

$$x=-\frac{28}{28}$$

Divide -28 by 28.

$$x=-1$$

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

$$14x^{2}+16x+2=14\left(x-\left(-\frac{1}{7}\right)\right)\left(x-\left(-1\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$14x^{2}+16x+2=14\left(x+\frac{1}{7}\right)\left(x+1\right)$$

Add \frac{1}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$14x^{2}+16x+2=14\times \frac{7x+1}{7}\left(x+1\right)$$

Cancel out 7, the greatest common factor in 14 and 7.

$$14x^{2}+16x+2=2\left(7x+1\right)\left(x+1\right)$$

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.This is achieved by dividing both sides of the equation by 14

$$x ^ 2 +\frac{8}{7}x +\frac{1}{7} = 0$$

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

$$r + s = -\frac{8}{7} rs = \frac{1}{7}$$

Two numbers r and s sum up to -\frac{8}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{7} = -\frac{4}{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>

$$r = -\frac{4}{7} - u s = -\frac{4}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{7}

$$(-\frac{4}{7} - u) (-\frac{4}{7} + u) = \frac{1}{7}$$

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

$$\frac{16}{49} - u^2 = \frac{1}{7}$$

Simplify the expression by subtracting \frac{16}{49} on both sides

$$-u^2 = \frac{1}{7}-\frac{16}{49} = -\frac{9}{49}$$

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

$$u^2 = \frac{9}{49} u = \pm\sqrt{\frac{9}{49}} = \pm \frac{3}{7}$$

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

$$r =-\frac{4}{7} - \frac{3}{7} = -1 s = -\frac{4}{7} + \frac{3}{7} = -0.143$$

Evaluate

$$2\left(x+1\right)\left(7x+1\right)$$