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

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Recognize Perfect Square Trinomial Structure: To determine the value of k that makes the given expression a perfect square trinomial, we need to recognize the structure of a perfect square trinomial. A perfect square trinomial is of the form (ax^2 + bx + c)^2, which expands to a^2x^4 + 2abx^3 + b^2x^2. However, since our given expression does not have an x^3 term, we are looking for a trinomial of the form (ax^2 + c)^2, which expands to a^2x^4 + 2acx^2 + c^2.Identify Perfect Square Terms: We can see that the first term of the given expression, 49x^4, is a perfect square, as it is (7x^2)^2. Similarly, the last term, 36y^4, is also a perfect square, as it is (6y^2)^2. For the expression to be a perfect square trinomial, the middle term -kx^2y^2 must be twice the product of the square roots of the first and last terms.Calculate Middle Term: The square root of the first term is 7x^2, and the square root of the last term is 6y^2. The product of these square roots is (7x^2)(6y^2) = 42x^2y^2. Since the middle term must be twice this product, we have -kx^2y^2 = 2 * 42x^2y^2.Solve for k: Solving for k, we get -kx^2y^2 = 84x^2y^2. This means that k must be -84 for the expression to be a perfect square trinomial.Final Value of k: Therefore, the value of k that makes 49x^4 - kx^2y^2 + 36y^4 a perfect square trinomial is k = -84.

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

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What is the value of kkk that makes 49x4−kx2y2+36y449x^{4}-kx^{2}y^{2}+36y^{4}49x4−kx2y2+36y4 a perfect square trinomial?

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