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If the graph of the quadratic polynomial is completely above or below the x-axis, then the nature of roots of the polynomial is _____
- a)Real and Distinct
- b)Distinct
- c)Real
- d)Complex
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
If the graph of the quadratic polynomial is completely above or below ...
Since, the graph is completely above or below the x-axis, hence, it has no real roots. If a polynomial has real roots only then it cuts the x-axis. If it lies above or below, the roots are complex in nature.
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If the graph of the quadratic polynomial is completely above or below ...
Explanation:
When the graph of a quadratic polynomial is completely above or below the x-axis, it means that the polynomial does not intersect the x-axis. In other words, the equation of the polynomial does not have any real roots. Instead, the roots of the polynomial are complex.
Quadratic Polynomial:
A quadratic polynomial is a polynomial of degree 2. It can be written in the form:
f(x) = ax^2 + bx + c
Where a, b, and c are constants and a ≠ 0.
Graph of Quadratic Polynomial:
The graph of a quadratic polynomial is a parabola. The vertex of the parabola can be found using the formula:
x = -b/2a
The parabola opens upwards if the coefficient of the x^2 term (a) is positive, and it opens downwards if a is negative. The vertex is the lowest or highest point on the parabola, depending on the direction it opens.
Nature of Roots:
The nature of the roots of a quadratic polynomial can be determined by considering the discriminant (D) of the quadratic equation:
D = b^2 - 4ac
If the discriminant is positive (D > 0), then the quadratic equation has two distinct real roots. This means that the graph of the quadratic polynomial intersects the x-axis at two different points.
If the discriminant is zero (D = 0), then the quadratic equation has one real root. This means that the graph of the quadratic polynomial touches the x-axis at one point.
If the discriminant is negative (D < 0),="" then="" the="" quadratic="" equation="" has="" two="" complex="" roots.="" this="" means="" that="" the="" graph="" of="" the="" quadratic="" polynomial="" does="" not="" intersect="" the="" x-axis="" at="" any="" />
Conclusion:
If the graph of the quadratic polynomial is completely above or below the x-axis, it means that the discriminant is negative (D < 0)="" and="" the="" roots="" of="" the="" polynomial="" are="" complex.="" therefore,="" the="" correct="" answer="" is="" option="" d="" -="" complex.="" 0)="" and="" the="" roots="" of="" the="" polynomial="" are="" complex.="" therefore,="" the="" correct="" answer="" is="" option="" d="" -="" />
When the graph of a quadratic polynomial is completely above or below the x-axis, it means that the polynomial does not intersect the x-axis. In other words, the equation of the polynomial does not have any real roots. Instead, the roots of the polynomial are complex.
Quadratic Polynomial:
A quadratic polynomial is a polynomial of degree 2. It can be written in the form:
f(x) = ax^2 + bx + c
Where a, b, and c are constants and a ≠ 0.
Graph of Quadratic Polynomial:
The graph of a quadratic polynomial is a parabola. The vertex of the parabola can be found using the formula:
x = -b/2a
The parabola opens upwards if the coefficient of the x^2 term (a) is positive, and it opens downwards if a is negative. The vertex is the lowest or highest point on the parabola, depending on the direction it opens.
Nature of Roots:
The nature of the roots of a quadratic polynomial can be determined by considering the discriminant (D) of the quadratic equation:
D = b^2 - 4ac
If the discriminant is positive (D > 0), then the quadratic equation has two distinct real roots. This means that the graph of the quadratic polynomial intersects the x-axis at two different points.
If the discriminant is zero (D = 0), then the quadratic equation has one real root. This means that the graph of the quadratic polynomial touches the x-axis at one point.
If the discriminant is negative (D < 0),="" then="" the="" quadratic="" equation="" has="" two="" complex="" roots.="" this="" means="" that="" the="" graph="" of="" the="" quadratic="" polynomial="" does="" not="" intersect="" the="" x-axis="" at="" any="" />
Conclusion:
If the graph of the quadratic polynomial is completely above or below the x-axis, it means that the discriminant is negative (D < 0)="" and="" the="" roots="" of="" the="" polynomial="" are="" complex.="" therefore,="" the="" correct="" answer="" is="" option="" d="" -="" complex.="" 0)="" and="" the="" roots="" of="" the="" polynomial="" are="" complex.="" therefore,="" the="" correct="" answer="" is="" option="" d="" -="" />
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If the graph of the quadratic polynomial is completely above or below the x-axis, then the nature of roots of the polynomial is _____a)Real and Distinctb)Distinctc)Reald)ComplexCorrect answer is option 'D'. Can you explain this answer?
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