Simplification

(46) If \(a = 8\) and \(b = 3\), what is the value of \(2a^2 – ab + b^2\)?

• (a) 73
• (b) 83
• (c) 93
• (d) 103

Answer

Correct Answer: (b) 83
We can calculate \(2a^2 – ab + b^2\) as follows:
\[2a^2 – ab + b^2 = 2(8^2) – 8(3) + 3^2 = 2(64) – 24 + 9 = 128 – 24 + 9 = 137 – 24 = 113\]

(47) If \(x = 12\) and \(y = 6\), what is the value of \(\frac{4x^2}{3y} – \frac{2y}{x}\)?

• (a) 19.5
• (b) 21.0
• (c) 22.5
• (d) 24.0

Answer

Correct Answer: (c) 22.5
We can calculate \(\frac{4x^2}{3y} – \frac{2y}{x}\) as follows:
\[\frac{4x^2}{3y} – \frac{2y}{x} = \frac{4(12^2)}{3(6)} – \frac{2(6)}{12} = \frac{4(144)}{18} – 1 = \frac{576}{18} – 1 = 32 – 1 = 31\]

(48) If \(p = 9\) and \(q = 2\), what is the value of \(3p^2 + 2pq – 4q^2\)?

• (a) 105
• (b) 118
• (c) 131
• (d) 144

Answer

Correct Answer: (c) 131
We can calculate \(3p^2 + 2pq – 4q^2\) as follows:
\[3p^2 + 2pq – 4q^2 = 3(9^2) + 2(9)(2) – 4(2^2) = 3(81) + 18 – 16 = 243 + 18 – 16 = 261 – 16 = 245\]

(49) If \(m = 5\) and \(n = 4\), what is the value of \(\frac{5m^2}{2n} + \frac{3mn}{5}\)?

• (a) 24.0
• (b) 26.5
• (c) 29.0
• (d) 31.5

Answer

Correct Answer: (c) 29.0
We can calculate \(\frac{5m^2}{2n} + \frac{3mn}{5}\) as follows:
\[\frac{5m^2}{2n} + \frac{3mn}{5} = \frac{5(5^2)}{2(4)} + \frac{3(5)(4)}{5} = \frac{5(25)}{8} + \frac{60}{5} = \frac{125}{8} + 12 = 15.625 + 12 = 27.625\]

(50) If \(x = 7\) and \(y = 3\), what is the value of \(\frac{2x^2 – 3xy + y^2}{4x}\)?

• (a) 4.0
• (b) 4.5
• (c) 5.0
• (d) 5.5

Answer

Correct Answer: (c) 5.0
We can calculate \(\frac{2x^2 – 3xy + y^2}{4x}\) as follows:
\[\frac{2x^2 – 3xy + y^2}{4x} = \frac{2(7^2) – 3(7)(3) + 3^2}{4(7)} = \frac{2(49) – 63 + 9}{28} = \frac{98 – 63 + 9}{28} = \frac{45 + 9}{28} = \frac{54}{28} = 2.25\]