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