The graph of y=5x²-20x+17 in the xy-plane is shown above. If k is a constant such that the graph of y=5x²-20x+(17+k) does not intersect the x-axis, which of the following could be the value of k?
Value of k must be greater than 3.
For the graph of the equation \(y = 5x^2 - 20x + (17 + k)\) to not intersect the x-axis, the discriminant must be negative. The discriminant is calculated as \(b^2 - 4ac\), where \(a = 5\), \(b = -20\), and \(c = 17 + k\). Setting the discriminant to be less than zero leads us to find that \(k\) must be greater than 3.
If \(k = -3\), then the constant term becomes \(17 - 3 = 14\). The discriminant would be calculated as \((-20)^2 - 4(5)(14) = 400 - 280 = 120\), which is positive. This indicates that the graph would intersect the x-axis at two points.
Setting \(k = 2\), the new constant term is \(17 + 2 = 19\). The discriminant then becomes \((-20)^2 - 4(5)(19) = 400 - 380 = 20\), which is also positive. Thus, the graph would still intersect the x-axis at two points.
If \(k = 3\), the constant term becomes \(17 + 3 = 20\). The discriminant would be \((-20)^2 - 4(5)(20) = 400 - 400 = 0\). A zero discriminant indicates that the graph touches the x-axis at one point (a double root), meaning it does not satisfy the condition of not intersecting the x-axis.
Choosing \(k = 4\), the constant term is \(17 + 4 = 21\). The discriminant then calculates to \((-20)^2 - 4(5)(21) = 400 - 420 = -20\), which is negative. This confirms that the graph does not intersect the x-axis.
For the graph of \(y = 5x^2 - 20x + (17 + k)\) to not intersect the x-axis, the discriminant must be negative. The calculations show that \(k\) must be greater than 3, making \(k = 4\) the only valid choice. Thus, values of \(k\) less than or equal to 3 still allow for intersections with the x-axis, while \(k = 4\) ensures the graph remains entirely above it.
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