Taking into account the definition of zeros of a function, the function has no real roots and its graph does not intersect the x-axis.
Zeros of a function
The function f(x) = ax² + bx + c
with a, b, c real numbers and a ≠ 0, is a function quadratic expressed in its polynomial form (It is so called because the function is expressed by a polynomial).
The points where a polynomial function crosses the axis of the independent term (x) represent the so-called zeros of the function.
That is, the zeros represent the roots of the polynomial equation that is obtained by making f(x)=0.
In summary, the roots or zeros of the quadratic function are those values of x for which the expression is equal to 0. Graphically, the roots correspond to the abscissa of the points where the parabola intersects the x-axis.
The following expression is called discriminant:
Δ= b²- 4×a×c
The discriminant determines the amount of roots of the function. Then:
- If Δ <0 the function has no real roots and its graph does not intersect the x-axis.
- If Δ> 0 the function has two real roots and its graph intersects the x-axis at two points .
- If Δ = 0 the function has a real root and its graph intersects the x-axis at a single point that coincides with its vertex. In this case the function is said to have a double root.
Amount of roots in this case
In this case, the function is f(x) = 3x² + 5x + 17, where:
Replacing in the definition of discriminant:
Δ= 5²- 4×3×17
and solving you get:
Δ= 25- 204
Δ= -179
Since Δ< 0, the function has no real roots and its graph does not intersect the x-axis.
Learn more about the zeros of a quadratic function:
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