Write the following equations in factored form. Remember to common factor first
(if possible).𝑦 = 𝑥^2 − 6𝑥 − 27
First, we can look for a common factor between the three terms. We notice that they are all divisible by 3, so we can factor out a 3:
𝑦 = 3(𝑥^2 − 6𝑥 − 9)
Next, we can try to factor the quadratic expression inside the parentheses. We can use the quadratic formula or complete the square to find that:
𝑥^2 − 6𝑥 − 9 = (𝑥 − 3)^2 − 18
Substituting this back into our original equation, we get:
𝑦 = 3((𝑥 − 3)^2 − 18)
Finally, we can distribute the 3 and write the equation in factored form:
𝑦 = 3(𝑥 − 3)^2 − 54
To factor the equation 𝑦 = 𝑥^2 - 6𝑥 - 27, let's first find the common factors, if any.
The quadratic expression 𝑥^2 - 6𝑥 - 27 cannot be factored further using common factors, so we will proceed with factoring it using the quadratic formula or completing the square.
Using the quadratic formula, which states that for any quadratic equation 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0, the solutions are given by:
𝑥 = (-𝑏 ± √(𝑏^2 - 4𝑎𝑐))/(2𝑎)
In our case, the equation is 𝑦 = 𝑥^2 - 6𝑥 - 27, so 𝑎 = 1, 𝑏 = -6, and 𝑐 = -27. Plugging these values into the quadratic formula, we get:
𝑥 = (6 ± √((-6)^2 - 4(1)(-27)))/(2(1))
Simplifying the expression under the square root:
𝑥 = (6 ± √(36 + 108))/2
𝑥 = (6 ± √144)/2
𝑥 = (6 ± 12)/2
So, the two solutions for 𝑥 are:
𝑥₁ = (6 + 12)/2 = 18/2 = 9
𝑥₂ = (6 - 12)/2 = -6/2 = -3
Therefore, the factored form of 𝑦 = 𝑥^2 - 6𝑥 - 27 is:
𝑦 = (𝑥 - 9)(𝑥 + 3)