3. If \( 4 x^{2}-12 x+k \) is a perfect square, 4. What is the value of \( d \) if \( 2 x-3 \) is a factor of \( 6 x^{2}+d x-12 \) ? 5. What must be added to \( (c+d)^{2} \) to get \( c^{2}+d^{2} \) ? 6. If \( x^{2}+\frac{1}{x^{2}}=11 \). What is the value of \( x+\frac{1}{x} \) ? 7. Simplify: \( \sqrt{9 x^{2}-12 x+4} \) 8. \( 9 p^{2}+A+25 q^{2} \) is a perfect square. What is the value(s) of

To determine if \( 4x^2 - 12x + k \) is a perfect square, we can relate it to the square of a binomial. For it to be a perfect square, the discriminant of the quadratic must be zero, allowing us to express it in the form \((ax + b)^2\). This gives \(k\) the value of \(9\), resulting in the expression being \((2x - 3)^2\). To find the value of \(d\) given that \(2x - 3\) is a factor of \(6x^2 + dx - 12\), we can equate \(6x^2 + dx - 12\) to \((2x - 3)(3x + m)\) where \(m\) is a value we need to determine. By matching coefficients, we can derive \(d\). Here, we find \(d = 6\). To transform \((c+d)^2\) into \(c^2 + d^2\), notice that \((c + d)^2 = c^2 + 2cd + d^2\). To isolate the needed addition, we simply subtract \(2cd\) from both sides, revealing that \(2cd\) must be added to \(c^2 + d^2\) to yield \((c + d)^2\). Using the relation \(x^2 + \frac{1}{x^2} = 11\), we can make an elegant substitution \(y = x + \frac{1}{x}\). By noticing that \(y^2 = x^2 + 2 + \frac{1}{x^2}\), we can manipulate the formula to find \(y\), thus solving for \(y^2 - 2 = 11\) leads us to conclude that \(y = 4\). For the simplification of \( \sqrt{9x^2 - 12x + 4} \), we can rewrite the expression as \( \sqrt{(3x - 2)^2}\). Recognizing the binomial square, we simplify it to \( |3x - 2| \). If \( 9p^2 + A + 25q^2 \) is a perfect square, we can think of it as a binomial squared, specifically \((3p + 5q + n)^2\). Expanding this form helps us identify that \(A\) corresponds to \(2n\) where \(n\) is the constant. To find specific values for \(A\), we can analyze possible combinations of terms, leading us to conclude that \(A\) should accommodate the necessary adjustments for the perfect square formation.