Solve 16x^2+19x+3

$$16 x ^ { 2 } + 19 x + 3$$

Factor

$$\left(x+1\right)\left(16x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 16x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=19 ab=16\times 3=48$$

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 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 19.

$$a=3 b=16$$

Rewrite 16x^{2}+19x+3 as \left(16x^{2}+3x\right)+\left(16x+3\right).

$$\left(16x^{2}+3x\right)+\left(16x+3\right)$$

Factor out x in 16x^{2}+3x.

$$x\left(16x+3\right)+16x+3$$

Factor out common term 16x+3 by using distributive property.

$$\left(16x+3\right)\left(x+1\right)$$

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.

$$16x^{2}+19x+3=0$$

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.

$$x=\frac{-19±\sqrt{19^{2}-4\times 16\times 3}}{2\times 16}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\times 16\times 3}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-19±\sqrt{361-64\times 3}}{2\times 16}$$

Multiply -64 times 3.

$$x=\frac{-19±\sqrt{361-192}}{2\times 16}$$

Add 361 to -192.

$$x=\frac{-19±\sqrt{169}}{2\times 16}$$

Take the square root of 169.

$$x=\frac{-19±13}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-19±13}{32}$$

Now solve the equation x=\frac{-19±13}{32} when ± is plus. Add -19 to 13.

$$x=-\frac{6}{32}$$

Reduce the fraction \frac{-6}{32} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{16}$$

Now solve the equation x=\frac{-19±13}{32} when ± is minus. Subtract 13 from -19.

$$x=-\frac{32}{32}$$

Divide -32 by 32.

$$x=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{16} for x_{1} and -1 for x_{2}.

$$16x^{2}+19x+3=16\left(x-\left(-\frac{3}{16}\right)\right)\left(x-\left(-1\right)\right)$$

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

$$16x^{2}+19x+3=16\left(x+\frac{3}{16}\right)\left(x+1\right)$$

Add \frac{3}{16} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$16x^{2}+19x+3=16\times \frac{16x+3}{16}\left(x+1\right)$$

Cancel out 16, the greatest common factor in 16 and 16.

$$16x^{2}+19x+3=\left(16x+3\right)\left(x+1\right)$$

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 16

$$x ^ 2 +\frac{19}{16}x +\frac{3}{16} = 0$$

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

$$r + s = -\frac{19}{16} rs = \frac{3}{16}$$

Two numbers r and s sum up to -\frac{19}{16} exactly when the average of the two numbers is \frac{1}{2}*-\frac{19}{16} = -\frac{19}{32}. 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>

$$r = -\frac{19}{32} - u s = -\frac{19}{32} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{16}

$$(-\frac{19}{32} - u) (-\frac{19}{32} + u) = \frac{3}{16}$$

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

$$\frac{361}{1024} - u^2 = \frac{3}{16}$$

Simplify the expression by subtracting \frac{361}{1024} on both sides

$$-u^2 = \frac{3}{16}-\frac{361}{1024} = -\frac{169}{1024}$$

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

$$u^2 = \frac{169}{1024} u = \pm\sqrt{\frac{169}{1024}} = \pm \frac{13}{32}$$

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

$$r =-\frac{19}{32} - \frac{13}{32} = -1 s = -\frac{19}{32} + \frac{13}{32} = -0.188$$

Evaluate

$$\left(x+1\right)\left(16x+3\right)$$