Find the quadratic polynomial if the sum of it's zeros is (1/4) and the product of the zeros is -1?

Question

Answer

Quadratic polynomial = x^2 - (alpha + beeta) x + alpha. beeta
= x^2 - (1/4)x + (-1)
= 4x^2 - x - 4 is your answer

Answer

Answer

Introduction:
To find the quadratic polynomial, we need to consider the sum and product of its zeros. A quadratic polynomial is in the form of ax^2 + bx + c, where a, b, and c are constants.

Given Information:
1. The sum of the zeros is 1/4.
2. The product of the zeros is -1.

Solution:
Let's consider the two zeros of the quadratic polynomial as x1 and x2.

Step 1: Sum of Zeros:
Given that the sum of the zeros is 1/4, we can write the equation as:
x1 + x2 = 1/4

Step 2: Product of Zeros:
Given that the product of the zeros is -1, we can write the equation as:
x1 * x2 = -1

Step 3: Expressing the Quadratic Polynomial:
To express the quadratic polynomial, we need to find the values of a, b, and c.

Since the quadratic polynomial is in the form of ax^2 + bx + c, we can use the following relationships:
- The sum of the zeros is given by: x1 + x2 = -b/a
- The product of the zeros is given by: x1 * x2 = c/a

Step 4: Solving for a, b, and c:
From Step 1, we have x1 + x2 = 1/4. Therefore, -b/a = 1/4.

From Step 2, we have x1 * x2 = -1. Therefore, c/a = -1.

Step 5: Solving for a:
From the equation -b/a = 1/4, we can rearrange it to: b/a = -1/4.
Since b/a is equal to -1/4, we can say b = -a/4.

Step 6: Solving for c:
From the equation c/a = -1, we can rearrange it to: c = -a.

Step 7: Expressing the Quadratic Polynomial:
Now that we have found the relationships between a, b, and c, we can express the quadratic polynomial as:
ax^2 + (-a/4)x + (-a)

Conclusion:
The quadratic polynomial, in this case, can be expressed as:
ax^2 - (a/4)x - a, where a is a constant.

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