The easiest marks on the paper, and the ones most often thrown away by a missing bracket.
Evaluate
Evaluate 3x² − 4xy + y² when x = −2 and y = 3.
3(−2)² = 12. The middle term is −4(−2)(3) = +24, because the two negatives cancel. And 3² = 9. Total: 12 + 24 + 9 = 45. B leaves the middle term at −24 — substitute in brackets and the signs handle themselves.
x
−2
y
3
Then replace each letter with the value in brackets — every time, even when it looks unnecessary.
3x² − 4xy + y²
the expression
3(−2)² − 4(−2)(3) + (3)²
substituted, nothing evaluated yet
The brackets are not decoration. They keep the minus sign attached to the 2, and they show that the 3 in front multiplies the whole square rather than sitting next to it.
3(−2)² = 3(4) = 12
square before multiplying
−4(−2)(3) = +24
two negatives, one positive
(3)² = 9
the last term
12 + 24 + 9 = 45
add them up
The middle term is −4xy, so with x negative it turns positive. Treating it as “subtract 24” gives 12 − 24 + 9 = −3, and that answer will be printed as a choice.
A power only ever attaches to the thing immediately in front of it. Without a bracket, that thing is the letter alone — never the minus sign.
C = 59(F − 32)
F is the one you were given
C = 59(95 − 32)
substitute
C = 59(63)
bracket first
C = 35
63 ÷ 9 = 7, × 5
Feeding 95 in as C and solving for F is a different question with a different answer. Underline which letter you were handed and which one is wanted before substituting anything.
Every copy takes the same value. Substitute them all in one pass.
f(x) = x² − 5x + 6, find f(−3)
two x's
f(−3) = (−3)² − 5(−3) + 6
both replaced
= 9 + 15 + 6
−5 × −3 = +15
= 30
done
For a² + b²a − b with a = 5, b = 3: work the top and the bottom completely before dividing — (25 + 9) ÷ (5 − 3) = 34 ÷ 2 = 17. A fraction bar is a bracket on both floors.
A rule such as a ◆ b = a² − 2b is just a formula with two slots. The first number goes into a, the second into b. Find (1 ◆ 2) ◆ 3.
1 ◆ 2 = (1)² − 2(2) = 1 − 4 = −3
the bracket first
(−3) ◆ 3 = (−3)² − 2(3)
the result becomes the new a
= 9 − 6 = 3
done
Swapping the slots changes the answer: 1 ◆ 2 = −3 but 2 ◆ 1 = 4 − 2 = 2. Never assume a made-up operation works both ways round — copy the rule exactly with the numbers in their own slots, in brackets.
Evaluate 4x² − 6x when x = −12.
4(−12)² = 4 × 14 = 1
square top and bottom; the sign goes
−6(−12) = +3
two negatives
1 + 3 = 4
done
Different: squaring a fraction makes it smaller — (½)² is ¼, not 1 or ½.
Evaluate x³ − x² when x = −3.
(−3)³ = −27
odd power: the sign survives
(−3)² = 9
even power: the sign goes
−27 − 9 = −36
subtract the whole 9
Different: a negative to an odd power stays negative, so the two terms do not cancel.
Treating (−3)³ like a square gives +27 and the answer 18. Count the minus signs: three of them multiply to a minus.
Evaluate x + 4x² − 9 when x = 3, then when x = −4.
x = 3: 79 − 9 = 7 ÷ 0
undefined
x = −4: 016 − 9 = 0 ÷ 7
= 0, perfectly fine
Different: zero on top gives 0; zero underneath gives no value at all.
Sometimes the “value” is itself an expression. It goes in exactly like a number — in brackets, into every copy of the letter. If f(x) = x² + 4x, find f(a + 2) − f(a).
f(a + 2) = (a + 2)² + 4(a + 2)
every x becomes (a + 2)
= a² + 4a + 4 + 4a + 8 = a² + 8a + 12
(a + 2)² has a middle term
f(a + 2) − f(a) = a² + 8a + 12 − (a² + 4a)
bracket the whole f(a)
= 4a + 12
test a = 1: f(3) − f(1) = 21 − 5 = 16 ✓
The reverse move: spot a block you already know. If 3x + y = 4, evaluate 9x² + 6xy + y² + 3x + y.
9x² + 6xy + y² = (3x + y)²
a perfect square in the block
expression = (3x + y)² + (3x + y)
rewrite in blocks
= 4² + 4 = 20
x and y never needed
Same idea when the given block is x − 1x = 3: square it. x² − 2(x)(1x) + 1x² = 9, and the middle term is just −2, so x² + 1x² = 11.
Squaring a block always produces a middle term. Dropping it — writing a² + 4, or forgetting the ±2 when squaring x ± 1x — is the trap these questions are built around.
Plain substitution items hide a sign trap (a negative value squared, or −4xy with x negative). The harder ones give a made-up operation or a block like 2x − y. Before expanding anything, ask whether the expression can be written in terms of the block you were given — it usually can, and that is the 10-second route.
3x² − 4xy + y² factors into (3x − y)(x − y). Same expression, so the same value.
Evaluate
Evaluate 2a² − 3ab − b² when a = −1 and b = 4.
2(−1)² = 2; −3(−1)(4) = +12; and −(4)² = −16. So 2 + 12 − 16 = −2. C makes the middle term −12. Note the last term is −(b²), not (−b)² — the minus sits outside the square.
