Powers and negative numbers · Quadratic Grand Prix
The mistake almost everyone makes here
One thing catches nearly every player out in the Quadratic Grand Prix. Once you understand it, you will not get it wrong again.
The small ² means multiply by itself
When you see x², it means x times x. That is all.
- x² = 3 × 39
- x² = 5 × 525
Easy so far. Now here is the part that trips people up.
Negative times negative is always positive
What if x = −3?
x² means x times x. So x² = (−3) × (−3).
Two negatives multiplied together always give a positive.
- (−3) × (−3)9
So x² = 9. Not −9.
Key point
Squaring a negative number always gives a positive answer. Always. No exceptions.
2x² + 1, with x = 3
- Swap x for 32 × 3² + 1
- Do the ² first — (3) × (3) = 92 × 9 + 1
- Multiply18 + 1
- Add19 ✓
2x² + 1, with x = −3
- Swap x for −32 × (−3)² + 1
- Do the ² first — (−3) × (−3) = 92 × 9 + 1
- Multiply18 + 1
- Add19 ✓
The answer is the same as with x = 3. Squaring removed the negative.
Order matters here too
In 2x², you must do x² before multiplying by 2.
With x = 3:
Wrong way
2 × 3 = 6 first, then 6² = 36.
Right way
3² = 9 first, then 2 × 9 = 18.
The ² always comes before the multiply.
x² − 1 ≥ 8, with x = −3
- Swap x for −3(−3)² − 1
- Do the ² first — (−3) × (−3) = 99 − 1
- Take away8
- Is 8 ≥ 8?Yes — 8 equals 8, and ≥ includes an exact match. ✓
Three easy mistakes
Mistake 1
Thinking (−3)² is −9
This is the most common mistake in the whole game.
(−3)² means (−3) × (−3). Two negatives. The answer is positive.
(−3) × (−3) = 9.
Mistake 2
Multiplying by 2 before doing the ²
In 2x² with x = 3, some people do 2 × 3 = 6 first, then 6² = 36. Wrong.
Do 3² = 9 first. Then 2 × 9 = 18.
Mistake 3
Forgetting the negative on x in other parts
Say the battery is x² + 2x and x = −3.
x² = 9. Good. But 2x = 2 × (−3) = −6. Not 2 × 3 = 6.
x is −3 everywhere. Do not drop the negative.