A quadratic equation is any equation that can be written as ax2 + bx + c = 0 with a ≠ 0. In the CDS & OTA Elementary Mathematics paper it is a guaranteed scorer — roots, the discriminant and sum-product relations appear almost every year. This Cavalier guide rebuilds the topic with clear methods, the key formulas and solved exam-style questions so you never lose these easy marks.
Why quadratic equations are a sure CDS scorer
The CDS Elementary Mathematics paper packs 100 questions into 120 minutes with negative marking, so you need topics that deliver fast, certain marks. Quadratic equations are exactly that. Almost every paper carries a cluster of questions on roots, the discriminant, or the sum and product of roots — and most can be solved in under a minute if you know the right tool.
The chapter also underpins later work in algebra, coordinate geometry and even physics-style word problems. A candidate who can spot whether a quadratic factorises, and who reaches for the discriminant when it does not, simply moves faster than one who blindly applies the formula every time.
Quadratics reward pattern recognition over brute force. Learn to read the equation first — does it factorise? Is it a perfect square? Only then pick your method.
Standard form and identifying a, b and c
A polynomial equation of degree 2 in one variable is a quadratic. Its general or standard form is:
ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0. Here a is the coefficient of x2, b the coefficient of x, and c the constant term.
The condition a ≠ 0 is essential — if a were 0 the equation would collapse to a linear one. A quadratic always has exactly two roots (also called zeros or solutions), though they may be equal or non-real.
Before doing anything, always rearrange the equation into standard form so that the right side is 0. For example, 2x2 = 5x − 3 must first become 2x2 − 5x + 3 = 0, giving a = 2, b = −5, c = 3. Many errors come from misreading the sign of b or c when the equation is left untidied.
Forgetting the sign when you move a term across the equals sign. In 2x2 = 5x − 3, b is −5 and c is +3, not +5 and −3. Always write the cleaned-up standard form before reading off a, b, c.
Method 1: Solving by factorisation
Factorisation is the fastest method when the roots are simple. The idea rests on the zero-product rule: if a product of two factors is 0, at least one of them must be 0.
To factorise ax2 + bx + c, split the middle term b into two parts whose product is a×c and whose sum is b.
To solve x2 + 7x + 12 = 0: find two numbers with product 12 and sum 7 → that is 3 and 4. Rewrite as x2 + 3x + 4x + 12 = 0 → x(x+3) + 4(x+3) = 0 → (x+3)(x+4) = 0. So x = −3 or x = −4.
The signs of the two numbers depend on the signs of b and c. If c is positive, both numbers share the sign of b; if c is negative, the two numbers carry opposite signs. Keeping this rule in mind makes the splitting almost mechanical.
Always try factorisation first when a, b, c are small whole numbers. It is faster than the formula and less error-prone. Switch to the formula only when no neat factor pair exists.
Method 2: Completing the square
When a quadratic does not factorise neatly, you can force it into a perfect-square form. This method also explains where the quadratic formula comes from.
The aim is to write the equation as (x + p)2 = q, then take square roots of both sides.
Solve x2 + 6x − 7 = 0 by completing the square.
The general half-coefficient trick — add (b÷2a)2 — is the engine of this method. You rarely need it in the exam because the formula is faster, but understanding it makes the discriminant feel natural rather than memorised.
Method 3: The quadratic formula
The quadratic formula solves every quadratic, including those that refuse to factorise. It is derived by completing the square on the general form ax2 + bx + c = 0.
x = [−b ± √(b2 − 4ac)] ÷ 2a. The quantity inside the square root, b2 − 4ac, is called the discriminant, written D or Δ.
Read off a, b and c from the standard form, substitute carefully (watch the signs!), and simplify. The ± gives you the two roots in one calculation.
Dropping a negative sign inside b2. Remember b2 is always positive — even if b = −5, b2 = 25. Also, the whole −b is in the numerator, so for b = −5 you write +5 there.
Because the formula never fails, it is your safe fallback. But it involves a square root and division, so reserve it for cases where factorisation is not obvious — that keeps your arithmetic light and fast.
The discriminant and the nature of roots
You can often answer a CDS question without finding the roots at all — just by computing the discriminant D = b2 − 4ac. Its sign tells you everything about the type of roots.
For ax2 + bx + c = 0 with real a, b, c:
- If D > 0 → two distinct real roots.
- If D = 0 → two equal (real, coincident) roots.
- If D < 0 → no real roots (the roots are complex/imaginary).
A further refinement: when D > 0 and D is a perfect square (and a, b, c are rational), the roots are rational; if D > 0 but not a perfect square, the roots are irrational and occur as conjugate pairs such as p ± √q.
When a question asks for the value of k that makes roots equal, set D = 0 and solve for k. This single condition answers a whole family of CDS questions in seconds.
Sum and product of roots
If α and β are the two roots of ax2 + bx + c = 0, you can find their sum and product directly from the coefficients — no need to solve the equation.
Sum of roots α + β = −b ÷ a. Product of roots α × β = c ÷ a.
These relations are gold for the CDS exam. They let you build equations, find one root from another, or evaluate symmetric expressions. To reconstruct a quadratic from its roots, use:
A quadratic with roots α and β is x2 − (sum)x + (product) = 0, i.e. x2 − (α+β)x + αβ = 0.
Many "find α2 + β2" questions are solved with the identity α2 + β2 = (α+β)2 − 2αβ. Substituting the sum and product gives the answer instantly, without ever computing the individual roots.
Worked example: combining the tools
For the equation 2x2 − 7x + 3 = 0, find the roots, the discriminant, and the value of α2 + β2.
Notice how the sum and product gave us α2 + β2 = 37÷4 without squaring the actual roots — a real time-saver under exam pressure.
Turning word problems into quadratics
CDS often hides a quadratic inside a word problem — on ages, areas, speeds or consecutive numbers. The skill is in setting up the equation; once you have ax2 + bx + c = 0, the methods above finish the job.
- Let the unknown be x and write each given condition as an algebraic statement.
- Form an equation, expand, and bring everything to one side.
- Solve, then reject any root that is impossible in context (a negative length, age or count).
Keeping a negative or fractional root that makes no physical sense. If x represents a number of people or a length, discard roots that are negative or not whole when the context demands it.
For example, "the product of two consecutive positive integers is 56" becomes x(x+1) = 56 → x2 + x − 56 = 0 → (x+8)(x−7) = 0. The roots are −8 and 7; only x = 7 fits "positive integers," so the numbers are 7 and 8.
Common mistakes that cost marks
Most quadratic errors are avoidable. Keep this checklist in mind.
- Reading a, b, c before rewriting the equation in standard form (right side = 0).
- Treating b2 as negative when b is negative — b2 is always positive.
- Using sum of roots = b÷a instead of −b÷a (the minus is easy to drop).
- Concluding "no roots" when D < 0 means no real roots — complex roots still exist.
- Forgetting to reject an inadmissible root in a word problem.
A quadratic always has two roots. Whether they are real, equal or complex is decided entirely by the discriminant. Check its sign before you commit to a method.
Previous-year style question
Q. If the equation x2 − kx + 9 = 0 has equal roots, what are the possible values of k?
Answer: For equal roots the discriminant must be zero. Here a = 1, b = −k, c = 9, so D = b2 − 4ac = k2 − 4(1)(9) = k2 − 36. Setting D = 0 gives k2 = 36, so k = +6 or −6.
Whenever a question mentions "equal roots," "coincident roots" or "a perfect-square quadratic," immediately write D = 0. It converts the problem into a simple equation in the unknown coefficient.
Quick revision
- Standard form: ax2 + bx + c = 0, a ≠ 0; always tidy to right-side-zero first.
- Solve by factorisation (split the middle term), completing the square, or the formula x = [−b ± √(b2−4ac)] ÷ 2a.
- Discriminant D = b2 − 4ac: D>0 distinct real, D=0 equal, D<0 no real roots; perfect-square D → rational roots.
- Sum of roots = −b÷a; product = c÷a; rebuild as x2 − (sum)x + (product) = 0.
- Identity α2+β2 = (α+β)2 − 2αβ answers symmetric questions fast.
Drill the discriminant and the sum-product rules until automatic. They let you answer most CDS quadratic questions without ever solving the equation in full.
Frequently asked questions
What makes an equation quadratic?
An equation is quadratic if it can be written as ax^2 + bx + c = 0 with a not equal to 0, that is, the highest power of the variable is 2. Such an equation always has exactly two roots.
How does the discriminant tell the nature of the roots?
Compute D = b^2 - 4ac. If D is greater than 0 there are two distinct real roots; if D equals 0 the roots are real and equal; if D is less than 0 there are no real roots (the roots are complex).
What are the sum and product of the roots in terms of coefficients?
For ax^2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. These let you analyse roots without actually solving the equation.
When should I factorise instead of using the quadratic formula?
Factorise first when a, b and c are small whole numbers and a neat factor pair is easy to spot, as it is faster and less error-prone. Use the formula when factorisation is not obvious or the roots are irrational.
How do I form a quadratic equation from its roots?
If the roots are alpha and beta, the equation is x^2 - (alpha + beta)x + (alpha x beta) = 0, that is x^2 - (sum of roots)x + (product of roots) = 0.
Can a quadratic equation have only one root?
Strictly, a quadratic always has two roots. When the discriminant is 0 the two roots are equal (coincident), so it looks like a single value but is counted as two equal roots.
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