Question
19. \( \left(x^{2}+\frac{3}{2}\right)\left(x^{2}-\frac{3}{2}\right) \)
Ask by Willis Deleon. in South Africa
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( x^{4}-\frac{9}{4} \).
Solution
Calculate or simplify the expression \( (x^2+\frac{3}{2})*(x^2-\frac{3}{2}) \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(x^{2}+\frac{3}{2}\right)\left(x^{2}-\frac{3}{2}\right)\)
- step1: Multiply the expression:
\(2^{-2}\left(4x^{4}-9\right)\)
- step2: Rearrange the terms:
\(x^{4}-\frac{9}{4}\)
The expression \( \left(x^{2}+\frac{3}{2}\right)\left(x^{2}-\frac{3}{2}\right) \) simplifies to \( x^{4}-\frac{9}{4} \).
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Simplify this solution Bonus Knowledge
To factor \( \left(x^{2}+\frac{3}{2}\right)\left(x^{2}-\frac{3}{2}\right) \), we can recognize that this expression is in the form of a difference of squares. The formula for the difference of squares is \( a^2 - b^2 = (a+b)(a-b) \). Here, we can let \( a = x^2 \) and \( b = \frac{3}{2} \). Thus, we have: \[ \left(x^{2}\right)^{2} - \left(\frac{3}{2}\right)^{2} = x^{4} - \frac{9}{4}. \] So, the factored form of the original expression is: \[ x^{4} - \frac{9}{4}. \] This can be rewritten as: \[ \left(x^2 + \frac{3}{2}\right)\left(x^2 - \frac{3}{2}\right) = x^{4} - \frac{9}{4}. \]
