A parabola is defined by the equation 𝑦 = 𝑥2 − 10𝑥 + 16.

Question

Write the equation in factored form

Answer 1

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.

Answer 2

y = (x-5)^2 - 9 is not considered "factored form"

you would want:
y = (x - 2)(x - 8)

Answer 3

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.

Answer 4

Geeeh bot, why not start with the original:

y = x^2 - 10x + 16
factor it to get

y = (x - 2)(x - 8)

Answer 5

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.