Solve 3n^2+7n+4

$$3 n ^ { 2 } + 7 n + 4$$

Factor

$$\left(n+1\right)\left(3n+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3n^{2}+an+bn+4. To find a and b, set up a system to be solved.

$$a+b=7 ab=3\times 4=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 7.

$$a=3 b=4$$

Rewrite 3n^{2}+7n+4 as \left(3n^{2}+3n\right)+\left(4n+4\right).

$$\left(3n^{2}+3n\right)+\left(4n+4\right)$$

Factor out 3n in the first and 4 in the second group.

$$3n\left(n+1\right)+4\left(n+1\right)$$

Factor out common term n+1 by using distributive property.

$$\left(n+1\right)\left(3n+4\right)$$

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.

$$3n^{2}+7n+4=0$$

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.

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

Square 7.

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

Multiply -4 times 3.

$$n=\frac{-7±\sqrt{49-12\times 4}}{2\times 3}$$

Multiply -12 times 4.

$$n=\frac{-7±\sqrt{49-48}}{2\times 3}$$

Add 49 to -48.

$$n=\frac{-7±\sqrt{1}}{2\times 3}$$

Take the square root of 1.

$$n=\frac{-7±1}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{-7±1}{6}$$

Now solve the equation n=\frac{-7±1}{6} when ± is plus. Add -7 to 1.

$$n=-\frac{6}{6}$$

Divide -6 by 6.

$$n=-1$$

Now solve the equation n=\frac{-7±1}{6} when ± is minus. Subtract 1 from -7.

$$n=-\frac{8}{6}$$

Reduce the fraction \frac{-8}{6} to lowest terms by extracting and canceling out 2.

$$n=-\frac{4}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -\frac{4}{3} for x_{2}.

$$3n^{2}+7n+4=3\left(n-\left(-1\right)\right)\left(n-\left(-\frac{4}{3}\right)\right)$$

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

$$3n^{2}+7n+4=3\left(n+1\right)\left(n+\frac{4}{3}\right)$$

Add \frac{4}{3} to n by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3n^{2}+7n+4=3\left(n+1\right)\times \frac{3n+4}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3n^{2}+7n+4=\left(n+1\right)\left(3n+4\right)$$

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

$$x ^ 2 +\frac{7}{3}x +\frac{4}{3} = 0$$

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

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

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{4}{3}

$$(-\frac{7}{6} - u) (-\frac{7}{6} + u) = \frac{4}{3}$$

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

$$\frac{49}{36} - u^2 = \frac{4}{3}$$

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

$$-u^2 = \frac{4}{3}-\frac{49}{36} = -\frac{1}{36}$$

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

$$u^2 = \frac{1}{36} u = \pm\sqrt{\frac{1}{36}} = \pm \frac{1}{6}$$

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

$$r =-\frac{7}{6} - \frac{1}{6} = -1.333 s = -\frac{7}{6} + \frac{1}{6} = -1.000$$

Evaluate

$$\left(n+1\right)\left(3n+4\right)$$