Solve 4g^2+11g+6

$$4 g ^ { 2 } + 11 g + 6$$

Factor

$$\left(g+2\right)\left(4g+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4g^{2}+ag+bg+6. To find a and b, set up a system to be solved.

$$a+b=11 ab=4\times 6=24$$

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 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 11.

$$a=3 b=8$$

Rewrite 4g^{2}+11g+6 as \left(4g^{2}+3g\right)+\left(8g+6\right).

$$\left(4g^{2}+3g\right)+\left(8g+6\right)$$

Factor out g in the first and 2 in the second group.

$$g\left(4g+3\right)+2\left(4g+3\right)$$

Factor out common term 4g+3 by using distributive property.

$$\left(4g+3\right)\left(g+2\right)$$

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.

$$4g^{2}+11g+6=0$$

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.

$$g=\frac{-11±\sqrt{11^{2}-4\times 4\times 6}}{2\times 4}$$

Square 11.

$$g=\frac{-11±\sqrt{121-4\times 4\times 6}}{2\times 4}$$

Multiply -4 times 4.

$$g=\frac{-11±\sqrt{121-16\times 6}}{2\times 4}$$

Multiply -16 times 6.

$$g=\frac{-11±\sqrt{121-96}}{2\times 4}$$

Add 121 to -96.

$$g=\frac{-11±\sqrt{25}}{2\times 4}$$

Take the square root of 25.

$$g=\frac{-11±5}{2\times 4}$$

Multiply 2 times 4.

$$g=\frac{-11±5}{8}$$

Now solve the equation g=\frac{-11±5}{8} when ± is plus. Add -11 to 5.

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

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

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

Now solve the equation g=\frac{-11±5}{8} when ± is minus. Subtract 5 from -11.

$$g=-\frac{16}{8}$$

Divide -16 by 8.

$$g=-2$$

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

$$4g^{2}+11g+6=4\left(g-\left(-\frac{3}{4}\right)\right)\left(g-\left(-2\right)\right)$$

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

$$4g^{2}+11g+6=4\left(g+\frac{3}{4}\right)\left(g+2\right)$$

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

$$4g^{2}+11g+6=4\times \frac{4g+3}{4}\left(g+2\right)$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4g^{2}+11g+6=\left(4g+3\right)\left(g+2\right)$$

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.This is achieved by dividing both sides of the equation by 4

$$x ^ 2 +\frac{11}{4}x +\frac{3}{2} = 0$$

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

$$r + s = -\frac{11}{4} rs = \frac{3}{2}$$

Two numbers r and s sum up to -\frac{11}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{11}{4} = -\frac{11}{8}. 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>

$$r = -\frac{11}{8} - u s = -\frac{11}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{2}

$$(-\frac{11}{8} - u) (-\frac{11}{8} + u) = \frac{3}{2}$$

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

$$\frac{121}{64} - u^2 = \frac{3}{2}$$

Simplify the expression by subtracting \frac{121}{64} on both sides

$$-u^2 = \frac{3}{2}-\frac{121}{64} = -\frac{25}{64}$$

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

$$u^2 = \frac{25}{64} u = \pm\sqrt{\frac{25}{64}} = \pm \frac{5}{8}$$

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

$$r =-\frac{11}{8} - \frac{5}{8} = -2 s = -\frac{11}{8} + \frac{5}{8} = -0.750$$

Evaluate

$$\left(g+2\right)\left(4g+3\right)$$