In order to factor 4x ^ 2 – 27x + 18 , we must break up the “bx” term into two terms whose coefficients…
add to-27 and multiply to 72.
add to 27 and multiply to 72.
add to 27 and multiply to 72.
add to 7 and multiply to -72.
The correct answer and explanation is:
To factor the quadratic expression 4×2−27x+184x^2 – 27x + 18, we need to break up the “bx” term into two parts. The goal is to find two numbers that add up to -27 (the coefficient of xx) and multiply to 72 (the product of 4×184 \times 18).
Step 1: Identify the product and sum.
The coefficient of x2x^2 is 4, and the constant term is 18. Therefore, the product of the first and last coefficients is: 4×18=724 \times 18 = 72
We need two numbers that add up to -27 and multiply to 72.
Step 2: Find two numbers that satisfy the conditions.
We need two numbers that add to -27 and multiply to 72. These numbers are -9 and -18 because: −9+(−18)=−27-9 + (-18) = -27
and −9×−18=72-9 \times -18 = 72
Step 3: Break up the “bx” term.
Now that we have the two numbers, we can split the middle term, −27x-27x, into −9x-9x and −18x-18x. This gives us: 4×2−9x−18x+184x^2 – 9x – 18x + 18
Step 4: Factor by grouping.
Next, we group the terms: (4×2−9x)−(18x−18)(4x^2 – 9x) – (18x – 18)
Now, factor out the greatest common factor (GCF) from each group: x(4x−9)−18(4x−9)x(4x – 9) – 18(4x – 9)
Step 5: Factor out the common binomial.
Since both terms contain (4x−9)(4x – 9), we can factor it out: (4x−9)(x−18)(4x – 9)(x – 18)
Thus, the factored form of 4×2−27x+184x^2 – 27x + 18 is: (4x−9)(x−18)(4x – 9)(x – 18)
The correct answer to the question is “add to -27 and multiply to 72.”