Class 8 Exam > Class 8 Questions > If(x2+3x+5)(x2−3x+5) = m2−n2, wha...
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If (x2+3x+5)(x2−3x+5) = m2−n2, what is the value of m?
- a)x2−3x
- b)3x
- c)x2+5
- d)x2+3x
Correct answer is option 'C'. Can you explain this answer?
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If(x2+3x+5)(x2−3x+5) = m2−n2, what is the value of m?a)x2&...
(x2+3x+5)(x2−3x+5) = (x2+5+3x)(x2+5−3x) = (x2+5)2−(3x)2 = m2−n2
∴ m = x2+5
∴ m = x2+5
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If(x2+3x+5)(x2−3x+5) = m2−n2, what is the value of m?a)x2&...
Understanding the Expression
To solve the equation (x² + 3x + 5)(x² - 3x + 5) = m² - n², we begin by recognizing that the left side is a product of two quadratic expressions.
Factoring the Expression
We can identify that the product can be expressed as a difference of squares:
m² - n² = (m - n)(m + n).
Thus, we need to find values for m and n such that the left side corresponds to this form.
Expanding the Left Side
Let's expand the left side:
1. First, multiply the two quadratics:
- x²(x²) = x^4
- x²(5) = 5x²
- 3x(x²) = 3x³
- 3x(-3x) = -9x²
- 3x(5) = 15x
- 5(-3x) = -15x
- 5(5) = 25
2. Combine like terms:
- x^4 + 3x³ + (5 - 9)x² + (15 - 15)x + 25
- This simplifies to: x^4 + 3x³ - 4x² + 25.
Identifying m
Now, we need to relate this to m² - n². The expression can be rearranged as m² = x^4 + 3x³ - 4x² + 25 + n².
To find m, we look for a suitable expression that fits the pattern. By inspection, we can test option C:
Testing Option C: m = x² + 5
If we set m = x² + 5, we find:
- m² = (x² + 5)² = x^4 + 10x² + 25.
This matches the remaining terms in our expanded expression. Therefore, m = x² + 5 is a valid choice.
Conclusion
Thus, the value of m in the equation is:
m = x² + 5, corresponding to option 'C'.
To solve the equation (x² + 3x + 5)(x² - 3x + 5) = m² - n², we begin by recognizing that the left side is a product of two quadratic expressions.
Factoring the Expression
We can identify that the product can be expressed as a difference of squares:
m² - n² = (m - n)(m + n).
Thus, we need to find values for m and n such that the left side corresponds to this form.
Expanding the Left Side
Let's expand the left side:
1. First, multiply the two quadratics:
- x²(x²) = x^4
- x²(5) = 5x²
- 3x(x²) = 3x³
- 3x(-3x) = -9x²
- 3x(5) = 15x
- 5(-3x) = -15x
- 5(5) = 25
2. Combine like terms:
- x^4 + 3x³ + (5 - 9)x² + (15 - 15)x + 25
- This simplifies to: x^4 + 3x³ - 4x² + 25.
Identifying m
Now, we need to relate this to m² - n². The expression can be rearranged as m² = x^4 + 3x³ - 4x² + 25 + n².
To find m, we look for a suitable expression that fits the pattern. By inspection, we can test option C:
Testing Option C: m = x² + 5
If we set m = x² + 5, we find:
- m² = (x² + 5)² = x^4 + 10x² + 25.
This matches the remaining terms in our expanded expression. Therefore, m = x² + 5 is a valid choice.
Conclusion
Thus, the value of m in the equation is:
m = x² + 5, corresponding to option 'C'.
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If(x2+3x+5)(x2−3x+5) = m2−n2, what is the value of m?a)x2−3xb)3xc)x2+5d)x2+3xCorrect answer is option 'C'. Can you explain this answer?
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