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Question Is the discriminant of the function yˉ=xˉ2‾+4‾xˉ+4‾\bar{y}=\bar{x} \overline{2}+\overline{4} \bar{x}+\overline{4} positive, negative, or zero?

Studdy Solution

STEP 1

Assumptions1. The given function is a quadratic function of the form yˉ=aˉxˉ+bˉxˉ+cˉ\bar{y}=\bar{a} \bar{x}^+\bar{b} \bar{x}+\bar{c}. The coefficients of the quadratic function are aˉ=xˉ,bˉ=4‾,cˉ=4‾\bar{a} = \bar{x}, \bar{b} = \overline{4}, \bar{c} = \overline{4}3. The discriminant of a quadratic function is given by the formula Δ=bˉ−4aˉcˉ\Delta = \bar{b}^ -4\bar{a}\bar{c}

STEP 2

First, we need to find the discriminant of the quadratic function. We can do this by substituting the values of a, b, and c into the formula for the discriminant.
Δ=bˉ2−4aˉcˉ\Delta = \bar{b}^2 -4\bar{a}\bar{c}

STEP 3

Now, plug in the given values for a, b, and c to calculate the discriminant.
Δ=‾2−xˉ‾\Delta = \overline{}^2 -\bar{x}\overline{}

STEP 4

Calculate the discriminant.
Δ=16−4xˉ4‾\Delta =16 -4\bar{x}\overline{4}

STEP 5

implify the discriminant.
Δ=16−16xˉ\Delta =16 -16\bar{x}

STEP 6

Now, we need to determine whether the discriminant is positive, negative, or zero. If the discriminant is positive, then the quadratic function has two distinct real roots. If the discriminant is zero, then the quadratic function has one real root (a repeated root). If the discriminant is negative, then the quadratic function has two complex roots (no real roots).
We can see that the discriminant is Δ=16−16xˉ\Delta =16 -16\bar{x}Since xˉ\bar{x} is the coefficient of xˉ2\bar{x}^2 in the quadratic function, it is a constant. Therefore, the value of the discriminant depends on the value of xˉ\bar{x}. If xˉ\bar{x} is less than1, then the discriminant is positive. If xˉ\bar{x} is equal to1, then the discriminant is zero. If xˉ\bar{x} is greater than1, then the discriminant is negative.
However, without knowing the value of xˉ\bar{x}, we cannot definitively say whether the discriminant is positive, negative, or zero. Therefore, the correct answer is "Not Sure".

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