A parabola is defined by the equation 𝑦 = 𝑥2 − 10𝑥 + 16.
Write the equation in factored form
To write the equation of the parabola in factored form, we need to use the method of completing the square.
First, let's group the quadratic terms:
𝑦 = 𝑥² - 10𝑥 + 16
Now we need to complete the square for the 𝑥 terms. To do that, we need to add and subtract a constant to the expression inside the parenthesis, where the constant is equal to half the coefficient of the 𝑥 term, squared.
The coefficient of the 𝑥 term is -10, so the constant we need to add and subtract is:
(-10/2)² = 25
Adding and subtracting 25 to the expression:
𝑦 = 𝑥² - 10𝑥 + 16 + 25 - 25
Now we can group the first three terms and write the last two terms as a perfect square:
𝑦 = (𝑥² - 10𝑥 + 25) + 16 - 25
The expression inside the parenthesis is a perfect square:
𝑦 = (𝑥 - 5)² - 9
So the factored form of the equation is:
𝑦 = (𝑥 - 5)² - 9
We can see that the vertex of the parabola is at (5,-9), and it opens upwards.
y = (x-5)^2 - 9 is not considered "factored form"
you would want:
y = (x - 2)(x - 8)
You are correct, I apologize for the mistake in my previous response.
To write the equation of the parabola in factored form, we can start with the completed square form:
𝑦 = (𝑥 - 5)² - 9
We can then factor the expression inside the parenthesis as a perfect square:
𝑦 = (𝑥 - 5 + 3)(𝑥 - 5 - 3)
Simplifying:
𝑦 = (𝑥 - 2)(𝑥 - 8)
So the factored form of the equation is indeed 𝑦 = (𝑥 - 2)(𝑥 - 8), which shows the two x-intercepts of the parabola at x = 2 and x = 8.
Thank you for bringing this to my attention.
Geeeh bot, why not start with the original:
y = x^2 - 10x + 16
factor it to get
y = (x - 2)(x - 8)
You are absolutely right, starting with the original equation and factoring it is a simpler and more straightforward approach.
So, to factor the original equation 𝑦 = 𝑥^2 − 10𝑥 + 16, we can use the methods of factoring quadratics.
We need to find two numbers whose product is 16 and whose sum is -10 (the coefficient of the 𝑥 term).
Those two numbers are -2 and -8.
So, we can write:
𝑦 = (𝑥 - 2)(𝑥 - 8)
This is the factored form of the equation, which shows the two x-intercepts of the parabola at x = 2 and x = 8.
Thank you for pointing this out and I apologize for any confusion caused by my previous response.
To write the equation 𝑦 = 𝑥^2 − 10𝑥 + 16 in factored form, we need to find the x-intercepts of the parabola.
The factored form of a quadratic equation is written as 𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠), where 𝑟 and 𝑠 are the x-intercepts (or roots) of the equation.
To find the x-intercepts, we set 𝑦 equal to zero and solve for 𝑥.
0 = 𝑥^2 − 10𝑥 + 16
Now, we can factor the quadratic equation 𝑥^2 − 10𝑥 + 16 to determine the roots:
0 = (𝑥 − 8)(𝑥 − 2)
Therefore, the factored form of the equation 𝑦 = 𝑥^2 − 10𝑥 + 16 is 𝑦 = (𝑥 − 8)(𝑥 − 2).
To write the equation of the parabola in factored form, we need to factorize the quadratic equation.
Step 1: Rewrite the equation in the form 𝑦 = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐.
The given equation 𝑦 = 𝑥^2 − 10𝑥 + 16 is already in this form with 𝑎 = 1, 𝑏 = -10, and 𝑐 = 16.
Step 2: Factorize the quadratic expression. To do this, we need to find two numbers whose product is equal to 𝑎𝑐 (the product of the coefficient of 𝑥^2 and the constant term), and whose sum is equal to 𝑏.
In this case, 𝑎 = 1, 𝑏 = -10, and 𝑐 = 16. So, we need to find two numbers whose product is equal to 1 * 16 = 16 and whose sum is equal to -10.
The numbers that meet these criteria are -2 and -8, since (-2) * (-8) = 16 and (-2) + (-8) = -10.
Step 3: Replace the middle term (-10𝑥) with the sum of the two numbers we found in Step 2. In this case, -2𝑥 and -8𝑥.
So, the equation becomes:
𝑦 = 𝑥^2 − 2𝑥 − 8𝑥 + 16
Step 4: Group the terms and factor by grouping. Group the first two terms and the last two terms:
(𝑥^2 − 2𝑥) + (-8𝑥 + 16)
Step 5: Factor out the common factor from each group:
𝑥(𝑥 − 2) - 8(𝑥 − 2)
Step 6: Notice that the terms (𝑥 − 2) are common, so factor it out:
(𝑥 − 2)(𝑥 - 8)
Therefore, the equation of the parabola in factored form is:
𝑦 = (𝑥 − 2)(𝑥 - 8)