A quadratic polynomial having zeroes - √5/sqrt2 and √5/sqrt2 is

The quadratic polynomial with given zeroes \(- \frac{\sqrt{5}}{\sqrt{2}}\) and \(\frac{\sqrt{5}}{\sqrt{2}}\) can be expressed using the fact that for a quadratic polynomial \(ax^2 + bx + c\), the sum of the roots (\(\alpha + \beta\)) and the product of the roots (\(\alpha \times \beta\)) are related to the coefficients as follows:

\[
\text{Sum of the roots} = -\frac{b}{a}
\]
\[
\text{Product of the roots} = \frac{c}{a}
\]

Given roots:
\[
\alpha = -\frac{\sqrt{5}}{\sqrt{2}}, \quad \beta = \frac{\sqrt{5}}{\sqrt{2}}
\]

1. **Sum of the roots**:
\[
\alpha + \beta = -\frac{\sqrt{5}}{\sqrt{2}} + \frac{\sqrt{5}}{\sqrt{2}} = 0
\]

2. **Product of the roots**:
\[
\alpha \times \beta = \left(-\frac{\sqrt{5}}{\sqrt{2}}\right) \times \frac{\sqrt{5}}{\sqrt{2}} = -\frac{5}{2}
\]

So, the quadratic polynomial can be written as:
\[
x^2 - (\text{Sum of the roots})x + (\text{Product of the roots}) = x^2 - 0x - \frac{5}{2} = x^2 - \frac{5}{2}
\]

To make the polynomial free of fractions, multiply by 2:
\[
2x^2 - 5
\]

Thus, the quadratic polynomial is:
\[
\boxed{2x^2 - 5}
\]

Would you like more details or have any questions? Here are some related questions to explore:

1. What is the general form of a quadratic equation?
2. How can you derive the sum and product of the roots for any quadratic equation?
3. What are the steps to solve a quadratic equation using the quadratic formula?
4. How do you factorize a quadratic polynomial?
5. What is the significance of the discriminant in a quadratic equation?

**Tip:** When given the roots of a quadratic polynomial, always remember to express the quadratic as \(a(x-\alpha)(x-\beta)\) for easier formulation.

Explore the quadratic polynomial whose roots are -√5/sqrt2 and √5/sqrt2. This detailed solution covers key algebra concepts like sum and product of roots and how to form the quadratic equation from given roots. Ideal for Grades 9-11 students.

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