Solve 4w^2+13w+3

$$4 w ^ { 2 } + 13 w + 3$$

Factor

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

Solution Steps

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

$$a+b=13 ab=4\times 3=12$$

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 13.

$$a=1 b=12$$

Rewrite 4w^{2}+13w+3 as \left(4w^{2}+w\right)+\left(12w+3\right).

$$\left(4w^{2}+w\right)+\left(12w+3\right)$$

Factor out w in the first and 3 in the second group.

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

Factor out common term 4w+1 by using distributive property.

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

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.

$$4w^{2}+13w+3=0$$

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.

$$w=\frac{-13±\sqrt{13^{2}-4\times 4\times 3}}{2\times 4}$$

Square 13.

$$w=\frac{-13±\sqrt{169-4\times 4\times 3}}{2\times 4}$$

Multiply -4 times 4.

$$w=\frac{-13±\sqrt{169-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$w=\frac{-13±\sqrt{169-48}}{2\times 4}$$

Add 169 to -48.

$$w=\frac{-13±\sqrt{121}}{2\times 4}$$

Take the square root of 121.

$$w=\frac{-13±11}{2\times 4}$$

Multiply 2 times 4.

$$w=\frac{-13±11}{8}$$

Now solve the equation w=\frac{-13±11}{8} when ± is plus. Add -13 to 11.

$$w=-\frac{2}{8}$$

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

$$w=-\frac{1}{4}$$

Now solve the equation w=\frac{-13±11}{8} when ± is minus. Subtract 11 from -13.

$$w=-\frac{24}{8}$$

Divide -24 by 8.

$$w=-3$$

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

$$4w^{2}+13w+3=4\left(w-\left(-\frac{1}{4}\right)\right)\left(w-\left(-3\right)\right)$$

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

$$4w^{2}+13w+3=4\left(w+\frac{1}{4}\right)\left(w+3\right)$$

Add \frac{1}{4} to w by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4w^{2}+13w+3=4\times \frac{4w+1}{4}\left(w+3\right)$$

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

$$4w^{2}+13w+3=\left(4w+1\right)\left(w+3\right)$$

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

$$x ^ 2 +\frac{13}{4}x +\frac{3}{4} = 0$$

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

$$r + s = -\frac{13}{4} rs = \frac{3}{4}$$

Two numbers r and s sum up to -\frac{13}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{4} = -\frac{13}{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>

$$r = -\frac{13}{8} - u s = -\frac{13}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{4}

$$(-\frac{13}{8} - u) (-\frac{13}{8} + u) = \frac{3}{4}$$

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

$$\frac{169}{64} - u^2 = \frac{3}{4}$$

Simplify the expression by subtracting \frac{169}{64} on both sides

$$-u^2 = \frac{3}{4}-\frac{169}{64} = -\frac{121}{64}$$

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

$$u^2 = \frac{121}{64} u = \pm\sqrt{\frac{121}{64}} = \pm \frac{11}{8}$$

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

$$r =-\frac{13}{8} - \frac{11}{8} = -3 s = -\frac{13}{8} + \frac{11}{8} = -0.250$$

Evaluate

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