Track quiz

Eight quick questions, one for each idea. Use the table if you get stuck.

Quadratic Equations — the decider and the methods it opens
Idea	Remember
The form	Tidy to `ax² + bx + c = 0`, with `a ≠ 0`.
The root	A value that balances both sides. One substitution tests it.
Split and solve	Split the middle, bracket, set each to 0. Dividing by x can lose the root 0.
The formula	`x = (−b ± √(b² − 4ac)) / 2a`, exact.
The decider	`b² − 4ac`: above zero two roots, zero one, below zero none.
Two kinds	Decider above zero, whole-number a, b, c: a perfect square gives rational roots. Any other gives irrational roots, still real.
Picking a method	Perfect square, 0 too: factorise. Above zero, not a square: formula. Below zero: none.
Stories	Name it, build it, solve it, keep the root the world allows.

The equation x² + 3x = 3x + 11 — which sentence names it correctly?

Linear, because the x terms on both sides match and cancel
Quadratic, because gathered it reads `x² − 11 = 0`
Not an equation, because both sides look equal

Which value is a root of 2x² − 5x − 3 = 0?

3, because `18 − 15 − 3 = 0`
4, because `32 − 20 − 3 = 9` is close
−3, because the equation carries minus signs

x² = 7x was solved by dividing, giving x = 7. What is the full solution?

x = 7 only — the dividing was legal
x = 0 and x = 7
x = 0 only, because 7 was already found

Solve 3x² − 5x + 2 = 0 by the formula.

`x = (5 ± 1)/6`, that is 1 and 2/3
`x = (5 ± 1)/3`, that is 2 and 4/3
`x = (−5 ± 1)/6`, that is −2/3 and −1

Without solving 4x² − 4x + 1 = 0, what can the decider already say?

One real value counted twice, because `16 − 16 = 0`
Two distinct real roots, because every quadratic has two
No real roots, because a is 4

The decider of x² − 4x + 1 = 0 is 12. What are its roots like?

No real roots, because 12 is not a perfect square number
Two irrational roots, because 12 is above 0, not a square
One repeated root, because the decider 12 is positive

For 2x² − 9x + 7 = 0, which method does the choosing rule pick, and why?

Neither — a positive decider means the roots are irrational
The formula, because a is not 1
Factorisation — the decider `81 − 56 = 25` is a perfect square

A guava orchard's length is 3 m more than its breadth. Its area is 40 m². Which breadth survives?

−8 m, because it is a root of the equation
8 m — the length, read as the breadth
5 m — from `b² + 3b − 40 = 0`, roots 5 and −8

Chapter 4 of your Mathematics book is Quadratic Equations. You can straighten an equation, split what splits, and read b² − 4ac. Its exercises will feel like the moves you just practised.

Next: Arithmetic Progressions — lists that step by the same amount. One sum saves you adding every term.

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