How to Find 'K' When Graph of Quadratic Polynomial Above X Axis?

Solving Polynomial Equations: Determining k for Graphs Above X-Axis. Hello viewers! Welcome to NumberX, and today we have an interesting question based on the properties of the graph of a quadratic polynomial. So let's dive right in!

The question is as follows: If the graph of the polynomial y = x^2 + kx - x + 9 is above the x-axis, what are the possible values of k?
We have four options to choose from, but we need to determine the correct one.

To solve this question, we need to understand what it means for a graph to be above the x-axis. It implies that the value of the polynomial, denoted as f(x), is always greater than zero throughout its entire domain.
In other words, the given quadratic polynomial has no real roots. And this leads us to the conclusion that the discriminant, denoted by capital "D," is negative.

We don't need to worry about the coefficient of x^2 because it's already positive. So let's focus on the rest of the expression.
The standard form of the given polynomial is x^2 + (k - 1)x + 9. Using the discriminant, we can set up an inequality.
The discriminant, D, is given by (k - 1)^2 - 4 * 1 * 9, which is less than zero.
Simplifying this inequality, we get (k - 1)^2 - 36 is negative.
Recognizing that 36 is equal to 6^2, we can rewrite the left-hand side of the inequality as
(k + 5)(k - 7); using the identity a-squared minus b-squared equals (a+b)(a-b).
Applying the wavy curve method, we can determine the values of k that satisfy this inequality and keep the graph above the x-axis.

After solving the inequality, we find that k lies in the open interval (-5, 7). This means that when k is within this interval, the graph will remain above the x-axis. Otherwise, it will either intersect the x-axis or lie below it.
So, after careful analysis, we can confidently say that option (B) is the correct answer for this particular question.
And that concludes our discussion for today. I hope you found it helpful and enjoyed diving into the properties of quadratic polynomials. If you have any more questions or topics you'd like us to cover, please let us know in the comments below.
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