To factor the quadratic expression \(3x^2 - 13x - 10\), you can follow these steps in order:
-
Identify the coefficients: For the quadratic \(ax^2 + bx + c\), identify \(a = 3\), \(b = -13\), and \(c = -10\).
-
Multiply \(a\) and \(c\): Calculate \(ac = 3 \times -10 = -30\).
-
Find two numbers that multiply to \(ac\) and add to \(b\): Look for two numbers that multiply to \(-30\) and add to \(-13\). The numbers \(-15\) and \(2\) work because \(-15 \times 2 = -30\) and \(-15 + 2 = -13\).
-
Rewrite the middle term: Rewrite the quadratic by breaking down the middle term using the two numbers found: \[ 3x^2 - 15x + 2x - 10 \]
-
Group terms: Group the terms into two pairs: \[ (3x^2 - 15x) + (2x - 10) \]
-
Factor out the common factors: Factor out the greatest common factor from each group: \[ 3x(x - 5) + 2(x - 5) \]
-
Factor out the common binomial: Notice that \(x - 5\) is a common factor: \[ (3x + 2)(x - 5) \]
-
Write the final factored form: The factored form of \(3x^2 - 13x - 10\) is: \[ (3x + 2)(x - 5) \]
These steps provide a clear pathway to factoring the given quadratic expression.