To factor the quadratic expression 4r² - 7r + 3, we can use the factoring method. We're looking for two binomials in the form (ar + b)(cr + d) where ac = 4, ad + bc = -7, and bd = 3. Here's how to factor it step by step:
Step 1: Multiply the leading coefficient (4) by the constant term (3). This gives us 4 * 3 = 12.
Step 2: Find two numbers that multiply to 12 (the result from step 1) and add up to the middle coefficient (-7). These numbers are -4 and -3 because (-4) * (-3) = 12, and (-4) + (-3) = -7.
Step 3: Rewrite the middle term (-7r) using the numbers from step 2 (-4 and -3). We split the middle term like this:
4r² - 4r - 3r + 3
Step 4: Group the terms and factor by grouping:
(4r² - 4r) - (3r - 3)
Step 5: Factor out the greatest common factor from each group:
4r(r - 1) - 3(r - 1)
Step 6: Now you can see that (r - 1) is a common factor in both terms. Factor it out:
(r - 1)(4r - 3)
So, the factored form of the quadratic expression 4r² - 7r + 3 is (r - 1)(4r - 3).