Question

analyzing equations use the discriminant to match the quadratic equation ( 2x^{2}-20x + 50 = 0 ) with the correct graph.
the discriminant is
. this means that the graph has
x - intercepts. the y - intercept is
.

Explanation:

Step1: Find the discriminant

For a quadratic equation \(ax^{2}+bx + c = 0\), the discriminant is \(\Delta=b^{2}-4ac\).
In the equation \(2x^{2}-20x + 50=0\), \(a = 2\), \(b=-20\), \(c = 50\).
So \(\Delta=(-20)^{2}-4\times2\times50\)
\(=400 - 400\)
\(=0\)

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

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

Step3: Find the \(y\) - intercept

To find the \(y\) - intercept, set \(x = 0\) in the equation \(y=2x^{2}-20x + 50\).
When \(x = 0\), \(y=2\times0^{2}-20\times0 + 50=50\)

Answer:

The discriminant is \(0\). This means that the graph has one \(x\) - intercept. The \(y\) - intercept is \(50\).