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Level 9Grade 9skill: expression_eval· 8 min read

Evaluating Algebraic Expressions

In short: To evaluate an expression, replace each variable with its value in brackets, then run BODMAS. The trap: -x² means -(x²), so at x = -3 it equals -9, not 9 — the exponent binds to the variable while the leading minus stays outside.

How to evaluate an expression

Evaluating an expression means turning letters into a single number. You are handed a value for each variable, you swap it in, and you compute. The whole game is won or lost in the swap: replace every variable with its value wrapped in brackets. Those brackets are invisible in the printed expression, but they are doing real work, and the moment a negative number shows up they are the difference between right and wrong.

Once the values are in, nothing new happens — you are back to ordinary arithmetic, so BODMAS runs the show: brackets, then powers, then multiplication and division left to right, then addition and subtraction. Substituting does not give you permission to read the line left to right like a sentence.

  1. Write down the value of each variable before you touch the expression.
  2. Replace every variable with its value in brackets — especially negatives, so -3 goes in as (-3).
  3. Resolve brackets and powers first: (-3)² is +9, because the bracket squares the sign too.
  4. Do multiplication and division next, working left to right.
  5. Do addition and subtraction last, working left to right.
  6. Read any lone minus in front of a term as '-1 times' — it applies to whatever the power or bracket produced, so resolve the power first and negate last. This is the single rule the whole sign trap turns on.

The one idea that kills the sign trap Why it works

-x² at x = -3
the exponent binds to x only; the front minus is -1 × (that square)

-(-3)² = -(9) = -9

(-x)² at x = -3
brackets pull the sign inside, so the whole thing is squared

(-(-3))² = (3)² = 9

5 - 2x² at x = -3
power first, then the product, then the subtraction

5 - 2(9) = 5 - 18 = -13

Here is the insight worth carrying out of this article: substitution is really a bracketing operation. When you write x = -3 into -x², the safe move is to picture -x² as -(x)² and drop the value straight into that inner bracket: -(-3)². Now the question 'is it +9 or -9?' answers itself, because you can see the exponent sitting on the bracket and the minus sitting outside it. The trap only bites people who substitute -3 as a loose, bracket-free -3 and then wonder which sign the square keeps.

Worked examples

Evaluate 5 - 2x² when x = -3. = -13
  1. Substitute with brackets: 5 - 2(-3)².
  2. Power first: (-3)² = 9.
  3. Product next: 2 × 9 = 18.
  4. Subtract last: 5 - 18.
Evaluate -x² + 3x when x = -5. = -40
  1. Read -x² as -(x)²: -(-5)² + 3(-5).
  2. Square inside the bracket: (-5)² = 25, so -(25) = -25.
  3. Multiply: 3 × (-5) = -15.
  4. Add: -25 + (-15).
Evaluate (a - b)² - ab when a = 4 and b = -2. = 44
  1. Substitute with brackets: (4 - (-2))² - (4)(-2).
  2. Simplify inside: 4 - (-2) = 6, so (6)².
  3. Power: 6² = 36; product: (4)(-2) = -8.
  4. Combine: 36 - (-8) = 36 + 8.
Evaluate 2x² - 3x + 1 when x = -2. = 15
  1. Substitute: 2(-2)² - 3(-2) + 1.
  2. Power first: (-2)² = 4, so 2 × 4 = 8.
  3. Product: 3 × (-2) = -6, and the sign in front makes it -(-6) = +6.
  4. Add across: 8 + 6 + 1.
Evaluate (x² - 9) / (x + 3) when x = -4. (Spot the shortcut, but check by substituting.) = -7
  1. The insight: x² - 9 factors as (x - 3)(x + 3), so the expression equals x - 3 wherever x ≠ -3 — at x = -4 that predicts -7.
  2. Verify by direct substitution: ((-4)² - 9) / ((-4) + 3).
  3. Top: 16 - 9 = 7. Bottom: -4 + 3 = -1.
  4. Divide: 7 / (-1).

Four ways strong students still lose the mark

A squared negative isn't negative — but the bare minus stays

✗ For -x² + 3x at x = -5, treating -x² as (-5)² = 25, giving 25 - 15 = 10.

The exponent in -x² attaches only to x; the minus in front means -1 × x². So -x² = -((-5)²) = -25, not +25. The squaring itself was fine — the slip was letting that leading -1 evaporate the instant the 25 showed up.

Fix: Rewrite -x² as -(x)² before substituting: -(-5)² = -(25) = -25. Then -25 + 3(-5) = -40.

sign_error

Substituting doesn't switch off BODMAS

✗ For 5 - 2x² at x = -3, reading left to right: 5 - 2 = 3, then 3 × 9 = 27.

After substitution you still owe powers before multiplication and multiplication before subtraction. Going left to right multiplies the leftover 3 by the 9 and scrambles the precedence.

Fix: Power first: (-3)² = 9. Then 2 × 9 = 18. Then 5 - 18 = -13.

order_of_ops_lr

x² means x·x, not 2x

✗ For 2x² - 3x + 1 at x = -2, reading x² as 2x: x² → 2(-2) = -4, giving 2(-4) + 6 + 1 = -1.

An exponent is repeated multiplication of the base by itself, so x² = (-2)(-2) = 4. Turning the exponent into a coefficient (×2) is the wrong operation entirely.

Fix: x² = (-2)(-2) = 4, so 2(4) - 3(-2) + 1 = 8 + 6 + 1 = 15.

wrong_operation

Finish the arithmetic — a half-done line is not an answer

✗ For 2x² - 3x + 1 at x = -2, doing the power and product to reach 8 + 6 + 1 but stopping at 8 + 6 = 14 (or handing in the unfinished line 8 + 6 + 1).

You did the hard part — the power and the product — then stalled a step short. 14 quietly drops the final + 1; the terms still have to be combined all the way down to a single number.

Fix: Carry it to the end: 8 + 6 + 1 = 15.

partial_computation

Practice: evaluate under pressure

Each problem gives you an expression and values to substitute. Bracket every value as you drop it in, run BODMAS honestly, and finish the arithmetic. Watch for the -x² versus (-x)² split and for lone minus signs — the tempting wrong answers you'll see are exactly the numbers you get by skipping a bracket or stopping early.

Before and after this skill

First, make sure of

Frequently asked questions

Why is -3² equal to -9 but (-3)² equal to 9?

In -3² the exponent attaches only to the 3, and the minus means -1 times that square: -(3²) = -(9) = -9. In (-3)² the brackets put the sign inside, so you square -3 itself: (-3)(-3) = 9. Same digits, different reach of the exponent.

When I substitute a negative value, do I really need brackets?

Yes, always. Writing the value in brackets keeps its sign glued to it and makes precedence obvious. Substituting x = -4 into x² as -4² invites the sign trap; writing (-4)² does not. Brackets cost nothing and prevent the most common lost mark on this topic.

Does x² mean x times 2?

No. x² means x multiplied by itself, x·x. At x = -2 that is (-2)(-2) = 4, not 2 × (-2) = -4. The small 2 is an exponent (how many times x is used as a factor), not a coefficient that multiplies x.

After I substitute, which operation do I do first?

Substituting does not change the order of operations. You still do brackets, then powers, then multiplication and division left to right, then addition and subtraction. Reading the line left to right like a sentence is the classic way to get a wrong answer.

How do I know what a minus sign in front of a term applies to?

A minus in front of a term means -1 times that whole term, applied after the power or bracket is resolved. So in -2x² you square x first, multiply by 2, then negate. Rewriting the term as -(2x²) before you substitute makes the reach of the minus unambiguous.