Questions: At how many points does the graph of the function below intersect the x axis? y=2x^2-7x+7 A. 1 B. 2 C. 0

Transcript text: At how many points does the graph of the function below intersect the $x$ axis? \[ y=2 x^{2}-7 x+7 \] A. 1 B. 2 C. 0

Solution

Solution Steps

Step 1: Identify the type of equation

The given equation is a quadratic function of the form $ y = ax^{2} + bx + c $, where $ a = 2 $, $ b = -7 $, and $ c = 7 $.

Step 2: Determine the discriminant

The discriminant ($ D $) of a quadratic equation $ ax^{2} + bx + c = 0 $ is given by: \[ D = b^{2} - 4ac \] Substitute the values of $ a $, $ b $, and $ c $: \[ D = (-7)^{2} - 4(2)(7) = 49 - 56 = -7 \]

Step 3: Analyze the discriminant

If $ D > 0 $, the quadratic equation has two distinct real roots, meaning the graph intersects the $ x $-axis at two points.
If $ D = 0 $, the quadratic equation has exactly one real root, meaning the graph touches the $ x $-axis at one point.
If $ D < 0 $, the quadratic equation has no real roots, meaning the graph does not intersect the $ x $-axis.

Since $ D = -7 < 0 $, the graph does not intersect the $ x $-axis.

Step 4: Select the correct option

The correct answer is C. 0.