Write the following equations in factored form. Remember to common factor first
(if possible).
𝑦 = 𝑥2 − 6𝑥 − 27
To factor 𝑦 = 𝑥2 − 6𝑥 − 27, we first look for common factors. 3 is a common factor of all three terms, since each term is divisible by 3. Therefore, we can factor out a 3:
𝑦 = 3(𝑥2 − 6𝑥 − 9)
Now we need to factor the quadratic expression inside the parentheses. To do this, we can use the quadratic formula, completing the square, or factoring by grouping. The quadratic can be factored as:
𝑦 = 3(𝑥 − 9)(𝑥 + 1)
Therefore, the factored form of 𝑦 = 𝑥2 − 6𝑥 − 27 is 𝑦 = 3(𝑥 − 9)(𝑥 + 1).
To factor the equation 𝑦 = 𝑥^2 − 6𝑥 − 27, we need to find two numbers that multiply to give -27 and add up to -6, because the equation is in the form of 𝑦 = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐.
Step 1: Find the factors of -27:
The factors of -27 are -1, 1, -3, 3, -9, and 9.
Step 2: Find the pair of factors that add up to -6:
After checking all the possible pairs, we find that -9 and 3 add up to -6.
Step 3: Rewrite the equation using the pair of factors:
𝑦 = (𝑥 − 9)(𝑥 + 3)
Therefore, the factored form of the equation 𝑦 = 𝑥^2 − 6𝑥 − 27 is 𝑦 = (𝑥 − 9)(𝑥 + 3).