IDNLearn.com is committed to providing high-quality answers to your questions. Ask your questions and receive comprehensive and trustworthy answers from our experienced community of professionals.
Sagot :
Let's determine which of the given expressions is a perfect square trinomial.
A perfect square trinomial takes the form \((a + b)^2\) or \((a - b)^2\), which expands to \(a^2 + 2ab + b^2\) for the addition form or \(a^2 - 2ab + b^2\) for the subtraction form.
Let's analyze each given expression:
1. \(50y^2 - 4x^2\)
This expression can be factored as a difference of squares:
[tex]\[50y^2 - 4x^2 = (5\sqrt{2}y)^2 - (2x)^2 = (5\sqrt{2}y - 2x)(5\sqrt{2}y + 2x)\][/tex]
This is not a perfect square trinomial.
2. \(100 - 36x^2y^2\)
This expression can also be factored as a difference of squares:
[tex]\[100 - 36x^2y^2 = (10)^2 - (6xy)^2 = (10 - 6xy)(10 + 6xy)\][/tex]
This too is not a perfect square trinomial.
3. \(16x^2 + 24xy + 9y^2\)
Let's factor this expression:
[tex]\[16x^2 + 24xy + 9y^2 = (4x + 3y)^2\][/tex]
[tex]\[ \text{Check the middle term: } 2ab = 2(4x)(3y) = 24xy \][/tex]
This expression matches the form \(a^2 + 2ab + b^2\), where \(a = 4x\) and \(b = 3y\).
Thus, \(16x^2 + 24xy + 9y^2\) is a perfect square trinomial.
4. \(49x^2 - 70xy + 10y^2\)
Consider whether this can be expressed as a square trinomial:
[tex]\[ (7x)^2 = 49x^2 \quad \text{and} \quad (c)^2 = 10y^2 \][/tex]
However, the middle term \(-70xy\) does not satisfy \(2ab \neq -70xy\).
Thus, \(49x^2 - 70xy + 10y^2\) is not a perfect square trinomial.
Based on the detailed analysis, the expression that is a perfect square trinomial is:
[tex]\[ 16x^2 + 24xy + 9y^2 \][/tex]
Therefore, the index of the perfect square trinomial is 3.
A perfect square trinomial takes the form \((a + b)^2\) or \((a - b)^2\), which expands to \(a^2 + 2ab + b^2\) for the addition form or \(a^2 - 2ab + b^2\) for the subtraction form.
Let's analyze each given expression:
1. \(50y^2 - 4x^2\)
This expression can be factored as a difference of squares:
[tex]\[50y^2 - 4x^2 = (5\sqrt{2}y)^2 - (2x)^2 = (5\sqrt{2}y - 2x)(5\sqrt{2}y + 2x)\][/tex]
This is not a perfect square trinomial.
2. \(100 - 36x^2y^2\)
This expression can also be factored as a difference of squares:
[tex]\[100 - 36x^2y^2 = (10)^2 - (6xy)^2 = (10 - 6xy)(10 + 6xy)\][/tex]
This too is not a perfect square trinomial.
3. \(16x^2 + 24xy + 9y^2\)
Let's factor this expression:
[tex]\[16x^2 + 24xy + 9y^2 = (4x + 3y)^2\][/tex]
[tex]\[ \text{Check the middle term: } 2ab = 2(4x)(3y) = 24xy \][/tex]
This expression matches the form \(a^2 + 2ab + b^2\), where \(a = 4x\) and \(b = 3y\).
Thus, \(16x^2 + 24xy + 9y^2\) is a perfect square trinomial.
4. \(49x^2 - 70xy + 10y^2\)
Consider whether this can be expressed as a square trinomial:
[tex]\[ (7x)^2 = 49x^2 \quad \text{and} \quad (c)^2 = 10y^2 \][/tex]
However, the middle term \(-70xy\) does not satisfy \(2ab \neq -70xy\).
Thus, \(49x^2 - 70xy + 10y^2\) is not a perfect square trinomial.
Based on the detailed analysis, the expression that is a perfect square trinomial is:
[tex]\[ 16x^2 + 24xy + 9y^2 \][/tex]
Therefore, the index of the perfect square trinomial is 3.
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.