Question

10n^2+3n-1. Factor

Answer 1

To factor the quadratic expression \(10n^2 + 3n - 1\), we can use the method of factoring by grouping, or we can employ the quadratic formula to find the roots and then express the quadratic in its factored form.

For a quadratic expression of the form \(ax^2 + bx + c\), where:

- \(a = 10\)
- \(b = 3\)
- \(c = -1\)

First, we can find two numbers that multiply to \(a \cdot c = 10 \cdot (-1) = -10\) and add to \(b = 3\).

The numbers that fit this are \(5\) and \(-2\), since \(5 \cdot (-2) = -10\) and \(5 + (-2) = 3\).

Now, we can rewrite the middle term using these two numbers:

\[
10n^2 + 5n - 2n - 1
\]

Next, we group the terms:

\[
(10n^2 + 5n) + (-2n - 1)
\]

Now we factor each group:

\[
5n(2n + 1) - 1(2n + 1)
\]

Notice that \((2n + 1)\) is a common factor:

\[
(2n + 1)(5n - 1)
\]

So, the factored form of \(10n^2 + 3n - 1\) is:

\[
(2n + 1)(5n - 1)
\]