Question

analyzing equations use the discriminant to match the quadratic equation ( x^{2}-6x + 25 = 0 ) with the correct graph of the related function. explain your reasoning. the discriminant is. this means that the graph has ( x )-intercept(s). the ( y )-intercept is.

Explanation:

Step1: Calculate the discriminant

For a quadratic equation \(ax^{2}+bx + c = 0\), the discriminant is \(\Delta=b^{2}-4ac\).
In the equation \(x^{2}-6x + 25=0\), \(a = 1\), \(b=-6\), \(c = 25\).
\(\Delta=(-6)^{2}-4\times1\times25=36 - 100=-64\)

Step2: Determine the number of \(x\) - intercepts

If \(\Delta>0\), the quadratic function has two \(x\) - intercepts; if \(\Delta = 0\), the quadratic function has one \(x\) - intercept; if \(\Delta<0\), the quadratic function has no real \(x\) - intercepts.
Since \(\Delta=-64<0\), the graph has no \(x\) - intercepts.

Step3: Find the \(y\) - intercept

For the quadratic function \(y=ax^{2}+bx + c\), the \(y\) - intercept is obtained by setting \(x = 0\).
When \(x = 0\), \(y=0^{2}-6\times0 + 25=25\)

Answer:

The discriminant is \(-64\). This means that the graph has \(no\) \(x\) - intercept(s). The \(y\) - intercept is \(25\).