Rewriting a quadratic function to find roots and vertex | Algebra I | Khan Academy | Summary and Q&A

TL;DR

Understanding different forms of a quadratic function helps to find its zeroes (x-intercepts) and the minimum point (vertex).

Key Insights

• 💁 Different forms of a quadratic function, such as factored form and completing the square form, can be used to find the zeroes and minimum point.
• 😑 Factoring the quadratic expression helps determine the x-intercepts or zeroes.
• ❎ Completing the square helps identify the vertex, which corresponds to the minimum value of the quadratic function.

Transcript

I have a function here defined as x squared minus 5x plus 6. And what I want us to think about is what other forms we can write this function in if we, say, wanted to find the 0s of this function. If we wanted to figure out where does this function intersect the x-axis, what form would we put this in? And then another form for maybe finding out wha... Read More

Questions & Answers

Q: How can the factored form of a quadratic function help find the zeroes of the function?

The factored form, like in the example of f(x) = (x-2)(x-3), shows that the function is equal to zero when either x-2 or x-3 is equal to zero. Therefore, the zeroes of the function are x = 2 and x = 3.

Q: What does completing the square help determine about a quadratic function?

Completing the square helps find the vertex (minimum point) of a parabola. In the example, f(x) = x^2 - 5x + 6 can be rewritten as f(x) = (x - 5/2)^2 - 1/4. The parabola reaches its minimum value at x = 5/2, which corresponds to the vertex (5/2, -1/4).

Q: Why is the quadratic expression always non-negative when using the completing the square form?

The square of any real number is always non-negative. So, when we square the quantity (x - 5/2), it can never become negative. The minimum value occurs when this square is equal to zero, which happens when x - 5/2 is equal to zero.

Q: What does the minimum value of a quadratic function represent?

The minimum value represents the lowest point of the parabola or the lowest value that the quadratic function can attain. In the context of the example, the minimum value is -1/4, which occurs at x = 5/2.

Summary & Key Takeaways

• The video discusses different forms of a quadratic function and how they can be used to find the zeroes and minimum point.

• The function f(x) = x^2 - 5x + 6 is used as an example.

• By factoring the quadratic expression, the zeroes of the function can be determined as x = 2 and x = 3.

• Completing the square is another method used to find the vertex of the parabola, which in this case is located at (5/2, -1/4).