Solve 16x^2-15x-1

$$16 x ^ { 2 } - 15 x - 1$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 16x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-15 ab=16\left(-1\right)=-16$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -15.

$$a=-16 b=1$$

Rewrite 16x^{2}-15x-1 as \left(16x^{2}-16x\right)+\left(x-1\right).

$$\left(16x^{2}-16x\right)+\left(x-1\right)$$

Factor out 16x in 16x^{2}-16x.

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

Factor out common term x-1 by using distributive property.

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

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.

$$16x^{2}-15x-1=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 16\left(-1\right)}}{2\times 16}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 16\left(-1\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-15\right)±\sqrt{225-64\left(-1\right)}}{2\times 16}$$

Multiply -64 times -1.

$$x=\frac{-\left(-15\right)±\sqrt{225+64}}{2\times 16}$$

Add 225 to 64.

$$x=\frac{-\left(-15\right)±\sqrt{289}}{2\times 16}$$

Take the square root of 289.

$$x=\frac{-\left(-15\right)±17}{2\times 16}$$

The opposite of -15 is 15.

$$x=\frac{15±17}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{15±17}{32}$$

Now solve the equation x=\frac{15±17}{32} when ± is plus. Add 15 to 17.

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

Divide 32 by 32.

$$x=1$$

Now solve the equation x=\frac{15±17}{32} when ± is minus. Subtract 17 from 15.

$$x=-\frac{2}{32}$$

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

$$x=-\frac{1}{16}$$

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{1}{16} for x_{2}.

$$16x^{2}-15x-1=16\left(x-1\right)\left(x-\left(-\frac{1}{16}\right)\right)$$

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

$$16x^{2}-15x-1=16\left(x-1\right)\left(x+\frac{1}{16}\right)$$

Add \frac{1}{16} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$16x^{2}-15x-1=16\left(x-1\right)\times \frac{16x+1}{16}$$

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

$$16x^{2}-15x-1=\left(x-1\right)\left(16x+1\right)$$

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

$$x ^ 2 -\frac{15}{16}x -\frac{1}{16} = 0$$

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

$$r + s = \frac{15}{16} rs = -\frac{1}{16}$$

Two numbers r and s sum up to \frac{15}{16} exactly when the average of the two numbers is \frac{1}{2}*\frac{15}{16} = \frac{15}{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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{15}{32} - u s = \frac{15}{32} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{16}

$$(\frac{15}{32} - u) (\frac{15}{32} + u) = -\frac{1}{16}$$

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

$$\frac{225}{1024} - u^2 = -\frac{1}{16}$$

Simplify the expression by subtracting \frac{225}{1024} on both sides

$$-u^2 = -\frac{1}{16}-\frac{225}{1024} = -\frac{289}{1024}$$

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

$$u^2 = \frac{289}{1024} u = \pm\sqrt{\frac{289}{1024}} = \pm \frac{17}{32}$$

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

$$r =\frac{15}{32} - \frac{17}{32} = -0.063 s = \frac{15}{32} + \frac{17}{32} = 1$$

Evaluate

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