Question

19 mark for review
9x² + 8 = nx
in the given equation, n is a constant. the equation has exactly one solution. what is the value of \\(\frac{n^2}{8}\\) ?
enter answer

Explanation:

Step1: Rearrange the equation

First, we rearrange the given equation \(9x^{2}+8 = nx\) into standard quadratic form \(ax^{2}+bx + c = 0\). Subtract \(nx\) from both sides to get \(9x^{2}-nx + 8=0\). Here, \(a = 9\), \(b=-n\), and \(c = 8\).

Step2: Use the discriminant for one solution

For a quadratic equation \(ax^{2}+bx + c = 0\) to have exactly one real solution, the discriminant \(\Delta=b^{2}-4ac\) must be equal to \(0\). Substitute \(a = 9\), \(b=-n\), and \(c = 8\) into the discriminant formula:

$$ (-n)^{2}-4\times9\times8=0 $$

Simplify the left - hand side:

$$ n^{2}-288 = 0 $$

Then, add 288 to both sides to solve for \(n^{2}\):

$$ n^{2}=288 $$

Step3: Calculate \(\frac{n^{2}}{8}\)

Now that we know \(n^{2}=288\), we substitute this value into the expression \(\frac{n^{2}}{8}\):

$$ \frac{n^{2}}{8}=\frac{288}{8} $$

Simplify the right - hand side: \(\frac{288}{8}=36\)

Answer:

36