Which of the following quadratic curves will not intersect with the x- axis?
Answer Details
A quadratic curve is a polynomial of degree 2, which means that it can be written in the form of \(y = ax^{2} + bx + c\), where a, b, and c are constants.
For a quadratic curve to intersect with the x-axis, it must have at least one x-intercept. An x-intercept is a point on the curve where the value of y is equal to zero. In other words, it is where the curve crosses the x-axis.
To find the x-intercepts of a quadratic curve, we need to solve the equation \(y = ax^{2} + bx + c\) for x when y = 0. This gives us the quadratic formula, which is:
$$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$
If the discriminant (\(b^{2} - 4ac\)) is greater than or equal to zero, then the quadratic curve will intersect with the x-axis. If the discriminant is less than zero, then the quadratic curve will not intersect with the x-axis.
Using this information, we can determine which of the given quadratic curves will not intersect with the x-axis:
- \(y = 2 - 4x - x^{2}\)
The discriminant for this quadratic is \(16 - 4(1)(2) = 8\), which is greater than zero. Therefore, this quadratic curve will intersect with the x-axis.
- \(y = x^{2} - 5x -1\)
The discriminant for this quadratic is \(25 + 4(1)(1) = 29\), which is greater than zero. Therefore, this quadratic curve will intersect with the x-axis.
- \(y = 2x^{2} - x - 1\)
The discriminant for this quadratic is \((-1)^{2} - 4(2)(-1) = 9\), which is greater than zero. Therefore, this quadratic curve will intersect with the x-axis.
- \(y = 3x^{2} - 2x + 4\)
The discriminant for this quadratic is \((-2)^{2} - 4(3)(4) = -44\), which is less than zero. Therefore, this quadratic curve will not intersect with the x-axis.
Therefore, the quadratic curve that will not intersect with the x-axis is the one described by the equation \(y = 3x^{2} - 2x + 4\).