Question

1. which of the following trinomials can be represented by a rectangle? (a) ( x^{2}+2 x + 14 ) (b) ( x^{2}+11 x + 30 ) (c) ( x^{2}+10 x + 3 ) (d) ( x^{2}+13 x + 5 )

Explanation:

Step1: Recall the discriminant formula

For a quadratic trinomial \(ax^{2}+bx + c\), the discriminant is \(\Delta=b^{2}-4ac\). If \(\Delta\) is a perfect square, the trinomial can be represented by a rectangle (i.e., can be factored).

Step2: Calculate discriminant for option (a)

For \(x^{2}+2x + 14\), \(a = 1\), \(b=2\), \(c = 14\). Then \(\Delta=(2)^{2}-4\times1\times14=4 - 56=- 52\).

Step3: Calculate discriminant for option (b)

For \(x^{2}+11x + 30\), \(a = 1\), \(b = 11\), \(c=30\). Then \(\Delta=(11)^{2}-4\times1\times30=121 - 120 = 1=1^{2}\).

Step4: Calculate discriminant for option (c)

For \(x^{2}+10x + 3\), \(a = 1\), \(b = 10\), \(c = 3\). Then \(\Delta=(10)^{2}-4\times1\times3=100 - 12 = 88\).

Step5: Calculate discriminant for option (d)

For \(x^{2}+13x + 5\), \(a = 1\), \(b = 13\), \(c = 5\). Then \(\Delta=(13)^{2}-4\times1\times5=169 - 20 = 149\).

Answer:

B. \(x^{2}+11x + 30\)