If the squared difference of the zeros of the quadratic polynomial f(x) = x² + px + 45 is equal to 144, find the value of p

Video Explanation

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Solution

Given polynomial:

f(x) = x² + px + 45

Step 1: Find α + β and αβ

Let the zeros of the polynomial be α and β.

Comparing f(x) = x² + px + 45 with ax² + bx + c:

a = 1,   b = p,   c = 45

α + β = −b/a = −p

αβ = c/a = 45

Step 2: Use the Formula for Squared Difference

(α − β)² = (α + β)² − 4αβ

= (−p)² − 4(45)

= p² − 180

Step 3: Use the Given Condition

According to the question:

(α − β)² = 144

∴ p² − 180 = 144

∴ p² = 324

∴ p = ±18

Final Answer

The value of p = 18 or p = −18.

Conclusion

Thus, if the squared difference of the zeros of the quadratic polynomial f(x) = x² + px + 45 is equal to 144, then the value of p is ±18.