Does a quadratic always have two roots?

Over the reals the discriminant decides: positive gives two distinct roots, zero a repeated root, negative none.

Why can I not divide both sides by x?

Dividing by x assumes x is not zero, which loses that root; rearrange and factorise instead.

S4 Mathematics: Quadratic Equations (Paper 2)

2026-10-31

Tutor Prep Guide · Secondary Mathematics

Ten original questions on the quadratic formula and the discriminant: solve x squared minus 5x plus 3 and 2x squared plus 7x minus 4 by formula, find the discriminant of x squared minus 4x plus 4, show that x squared plus x plus 1 has no real roots, find k when x squared plus kx plus 9 has a repeated root, and solve 3x squared minus 5x minus 2. The challenge items turn a sum and product into a quadratic and complete the square for x squared plus 8x plus 5, and the paper ends with the error of claiming real roots from a negative discriminant.

Direct answer　Compute the discriminant first: positive gives two roots, zero a repeated root, negative no real roots.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S4 Mathematics: Quadratic Equations
Level	Four basic, three advanced, two challenge and one error-repair item

02　Questions (10 items)

FoundationUse the quadratic formula to solve x squared minus 5x plus 3 equals zero.
FoundationSolve 2x squared plus 7x minus 4 equals zero.
FoundationFind the discriminant of x squared minus 4x plus 4 equals zero and state how many distinct real roots it has.
FoundationSolve x squared plus 2x minus 8 equals zero.
IntermediateDoes x squared plus x plus 1 equals zero have real roots? Give a reason.
IntermediateIf x squared plus kx plus 9 equals zero has a repeated root, find the positive value of k.
IntermediateSolve 3x squared minus 5x minus 2 equals zero.
ChallengeTwo numbers add to 11 and multiply to 30. Find them.
ChallengeWrite x squared plus 8x plus 5 in the form (x plus p) squared plus q and state its minimum value.
Error redoError repair: a student obtains a discriminant of minus 9 but still writes two real roots. What is the correct conclusion?

03　Answers and one-line notes

#	Answer	One-line note
1	x equals (5 plus root 13) over 2 or (5 minus root 13) over 2	The discriminant is thirteen, so the formula gives five plus or minus root thirteen over two.
2	x equals one half or x equals minus 4	The discriminant is eighty-one, whose root is nine, giving one half and minus four.
3	Discriminant 0, one repeated root x equals 2	The discriminant is zero, which means the two roots coincide at two.
4	x equals 2 or x equals minus 4	The factors are x minus two and x plus four.
5	No real roots, because the discriminant is minus 3	The discriminant is minus three, which is negative, so there are no real roots.
6	k equals 6	A repeated root means the discriminant is zero, so k squared is thirty-six and the positive value is six.
7	x equals 2 or x equals minus one third	The discriminant is forty-nine, giving two and minus one third.
8	5 and 6	Treat the numbers as roots of t squared minus 11t plus 30, which factorises as t minus five times t minus six.
9	p equals 4 and q equals minus 11, minimum value minus 11	Half of eight is four, and adding and subtracting sixteen gives a minimum of minus eleven.
10	The discriminant is negative, so there are no real roots	A negative discriminant means the curve never meets the x axis, so there are no real roots.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.