Quadratic Equations in CAT: 5 Root Shortcuts, No Formula

Quant

Most aspirants meet a quadratic equation in CAT and reach for the formula. They write down the discriminant, take the square root, and spend ninety seconds on a question whose answer was sitting in the coefficients the whole time. The formula always works. It is rarely the fastest way in.

Quadratic questions in CAT Quant are usually not asking for the roots at all. They ask for a sum, a product, a condition on a constant, or the sign of a root. This guide gives you five root properties that answer those questions straight from the coefficients, so you only solve for the roots when you have no other choice.

Why the Formula Is the Slow Route in CAT Quadratics

Take a standard equation ax²+bx+c=0 with roots α and β. The quadratic formula gives you both roots in one go, but it also hides the structure that most CAT questions are built on. The sum of the roots is -b/a and the product is c/a. Those two numbers carry almost everything the question can legitimately ask.

That is why the questions look unfriendly. A typical one reads: the roots of x²-6x+4=0 are α and β, find α²+β². If you solve for the roots you land on 3 plus or minus the square root of 5, and you must now square and add two irrational numbers. If you read the sum as 6 and the product as 4, the answer is 36 minus 8, which is 28, in about ten seconds.

The aim is not to avoid algebra. It is to recognise which questions are about the relationship between the roots and which are about the roots themselves. The first kind has a shortcut. The second kind is where you factorise or use the formula.

The Root Property Check: Five Shortcuts in Order

The Root Property Check is a short sequence you run on any quadratic before you commit to solving it. You do not need all five every time. Most questions are settled by the first two or three.

The five checks are ordered by how often they pay off. Reading the sum and product costs nothing. The coefficient pattern costs three seconds of mental addition. By the time you reach the ratio condition, you have usually already seen what the question wants.

Shortcut One and Two: Sum, Product and the Free Root

For ax²+bx+c=0, the sum of the roots is -b/a and the product is c/a. In x²-7x+12=0, the sum is 7 and the product is 12, so the roots are 3 and 4 without any formula. You can also run the logic backwards. If a question says the roots add to 7 and multiply to 12, the equation is x²-7x+12=0.

The second shortcut is a coefficient pattern. If the three coefficients add to zero, then x=1 is a root, and the other root is c/a. In 3x²-8x+5=0, the coefficients 3, -8 and 5 add to zero, so 1 is a root and the other root is 5/3. Similarly, if a-b+c=0, then x=-1 is a root and the other is -c/a.

These two checks together dispose of a surprising number of questions, especially those with a constant like k in the equation. If k appears in one coefficient and you are told that 1 is a root, substitute x=1 and you get k immediately. You never expand anything.

Shortcut Three: Symmetric Expressions Without the Roots

A symmetric expression is one that does not change if you swap α and β. The sum of squares, the sum of cubes and the sum of reciprocals are all symmetric, and each can be written using only S, the sum, and P, the product.

The three you need are worth memorising. The sum of squares α²+β² equals S²-2P. The sum of cubes α³+β³ equals S³-3SP. The sum of reciprocals 1/α+1/β equals S/P. For x²-6x+4=0, S is 6 and P is 4, so the sum of squares is 28, the sum of cubes is 216-72=144, and the sum of reciprocals is 6/4=1.5.

The difference of the roots is also available. The square of the difference, (α-β)², equals S²-4P, which is the discriminant divided by a². So the gap between the roots is the square root of the discriminant over a. When a question asks how far apart two roots are, you read it off the coefficients.

Expression	In terms of S and P	Value for x²-6x+4=0
α²+β²	S²-2P	28
α³+β³	S³-3SP	144
1/α+1/β	S/P	1.5
(α-β)²	S²-4P	20

The last row gives 36 minus 16, which is 20, so the roots differ by the square root of 20. You can confirm it against the roots 3 plus or minus the square root of 5, whose gap is twice the square root of 5, and that squares to 20. The shortcut and the long route agree, which is the check you should run on your first few practice questions.

Shortcut Four and Five: Ratio of Roots and Sign of Roots

Many CAT quadratics hide the condition in words. One root is twice the other. The roots are reciprocals. The roots are equal in size and opposite in sign. Each of these has a single equation in the coefficients, so you can solve for the unknown constant without finding either root.

If the roots are in the ratio k to 1, the condition is k·b²=(k+1)²·a·c. Take x²-9x+k=0 with one root twice the other. Here k is 2, b is -9, a is 1, and the constant is the unknown. So 2·81=9·c, which gives c=18. The roots are 6 and 3, which add to 9 and multiply to 18, as they should.

The special cases are easier still. If the roots are reciprocals, then a=c. If the roots are equal and opposite, then b=0. If one root is zero, then c=0. These come directly from the sum and product, and each saves you a page of working.

Sign questions use three facts together. Real roots need a non-negative discriminant. Both roots positive need the sum and product positive. Both roots negative need a positive product and a negative sum. For x²-5x+k=0 to have two positive real roots, the product k must be positive and the discriminant 25-4k must be at least zero, so k lies between 0 and 6.25.

Condition on the roots	What it means for the coefficients
Roots are reciprocals	a equals c
Roots equal and opposite	b equals 0
One root is zero	c equals 0
Both roots positive and real	S>0, P>0 and D≥0
Roots in ratio k to 1	k·b²=(k+1)²·a·c

When to Solve for the Roots Anyway

Shortcuts are not a religion. Some questions ask for a specific root, or for the number of integer solutions, or give a quadratic inside a larger expression where the roots genuinely matter. For these you factorise if the coefficients are small, or use the formula if they are not.

A quick test is to look at the discriminant. If it is a perfect square, the roots are rational and factorising is fast. If it is not, the roots are irrational and the symmetric expressions are almost always the better route. That single check, taking a few seconds, tells you which way to go.

Integer-root questions deserve a separate note. If the roots must be integers, the product c/a restricts the pairs you can try, and the sum -b/a picks the pair. You list the factor pairs of the product and see which one adds to the sum. That is faster than any formula, and it works because you are using the same two numbers as before.

The algebraic structure here connects to other chapters. Our guide on algebraic versus numerical methods in CAT Quant explains how to decide between a clean derivation and plugging in values, and our longer list of quadratic formulas with previous year questions is a good reference once the five checks feel natural.

Turn the Five Checks into a Habit

Reading about shortcuts does little. The habit forms when you tag each practice question with the check that would have solved it fastest. After twenty or thirty questions, the tags show you which checks you reach for and which you forget. Most aspirants forget the ratio condition and the sign conditions.

Build a short list of the five checks and keep it next to your practice sheet, or save it in your CAT formula sheets and cheatsheets. When you review a wrong answer, ask which check you skipped. If your accuracy on algebra is uneven across the week, a CAT study planner can schedule short, repeated sessions on this chapter so the checks stay fresh.

Do not use the checks as a reason to avoid solving. On some days the quickest answer will be to factorise on sight. The point is to choose on purpose, and to know that you have options when the numbers look ugly.