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What is the value of k that makes 49x^(4)-kx^(2)y^(2)+36y^(4) a perfect square trinomial?

What is the value of k k that makes 49x4−kx2y2+36y4 49 x^{4}-k x^{2} y^{2}+36 y^{4} a perfect square trinomial?

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Full solution

Q. What is the value of k k that makes 49x4−kx2y2+36y4 49 x^{4}-k x^{2} y^{2}+36 y^{4} a perfect square trinomial?
  1. Identify Perfect Square Trinomial: To make 49x4−kx2y2+36y449x^{4}-kx^{2}y^{2}+36y^{4} a perfect square trinomial, the middle term coefficient kk must be such that the trinomial can be factored into (ax2+by2)2.(ax^{2} + by^{2})^{2}.
  2. Recognize Perfect Squares: The first term 49x449x^{4} is a perfect square, (7x2)2(7x^2)^2, and the last term 36y436y^{4} is a perfect square, (6y2)2(6y^2)^2.
  3. Calculate Middle Term Coefficient: For the trinomial to be a perfect square, the middle term coefficient kk must be 22 times the product of the square roots of the first and last terms, so k=2×7x2×6y2k = 2 \times 7x^2 \times 6y^2.
  4. Determine k Value: Calculate k: k=2×7×6×x2×y2k = 2 \times 7 \times 6 \times x^2 \times y^2.
  5. Determine k Value: Calculate k: k=2×7×6×x2×y2k = 2 \times 7 \times 6 \times x^2 \times y^2.k=84x2y2k = 84x^2y^2.

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