Solve 3x^2+7x+1

$$3 x ^ { 2 } + 7 x + 1$$

Factor

$$3\left(x-\frac{-\sqrt{37}-7}{6}\right)\left(x-\frac{\sqrt{37}-7}{6}\right)$$

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.

$$3x^{2}+7x+1=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 3}}{2\times 3}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-7±\sqrt{49-12}}{2\times 3}$$

Add 49 to -12.

$$x=\frac{-7±\sqrt{37}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-7±\sqrt{37}}{6}$$

Now solve the equation x=\frac{-7±\sqrt{37}}{6} when ± is plus. Add -7 to \sqrt{37}.

$$x=\frac{\sqrt{37}-7}{6}$$

Now solve the equation x=\frac{-7±\sqrt{37}}{6} when ± is minus. Subtract \sqrt{37} from -7.

$$x=\frac{-\sqrt{37}-7}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+\sqrt{37}}{6} for x_{1} and \frac{-7-\sqrt{37}}{6} for x_{2}.

$$3x^{2}+7x+1=3\left(x-\frac{\sqrt{37}-7}{6}\right)\left(x-\frac{-\sqrt{37}-7}{6}\right)$$

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 3

$$x ^ 2 +\frac{7}{3}x +\frac{1}{3} = 0$$

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

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

Two numbers r and s sum up to -\frac{7}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{3} = -\frac{7}{6}. 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>

$$r = -\frac{7}{6} - u s = -\frac{7}{6} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{3}

$$(-\frac{7}{6} - u) (-\frac{7}{6} + u) = \frac{1}{3}$$

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

$$\frac{49}{36} - u^2 = \frac{1}{3}$$

Simplify the expression by subtracting \frac{49}{36} on both sides

$$-u^2 = \frac{1}{3}-\frac{49}{36} = -\frac{37}{36}$$

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

$$u^2 = \frac{37}{36} u = \pm\sqrt{\frac{37}{36}} = \pm \frac{\sqrt{37}}{6}$$

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

$$r =-\frac{7}{6} - \frac{\sqrt{37}}{6} = -2.180 s = -\frac{7}{6} + \frac{\sqrt{37}}{6} = -0.153$$

Evaluate

$$3x^{2}+7x+1$$