Quadratic equations: compare the roots without solving twice

The quadratic set in a bank exam is three to five questions, and it is unusual in that the question is not really about quadratics. You are given two equations, one in x and one in y, and asked only for the relationship between them: is x greater than y, less than y, greater than or equal, less than or equal, or can no relationship be established. The roots are a means to that comparison and nothing more.

Candidates lose these marks not in factorising, which most can do, but in the comparison step, which most do carelessly. The overlap case in particular is designed to catch anyone who compares only one root against one root.

Factorise to roots, but keep the goal in view

Both equations are standard quadratics, and the fastest route to the roots is the sum-and- product split you already use: for x² − 7x + 12, find two numbers that multiply to 12 and add to 7, giving roots 3 and 4. Do this for both equations and you have two roots for x and two for y.

The trap begins here. With two values of x and two of y, there are four pairwise comparisons, and the relationship only holds if it holds for all four. Comparing the larger x to the larger y and stopping is the single most common error in the topic.

The sign shortcut for a fast start

You can often read the signs of the roots straight from the equation without solving, which tells you immediately whether a comparison is even going to be clean.

For a quadratic written as x² + bx + c:

• If c is positive, both roots share a sign; that shared sign is opposite to the sign of b. So x² + 7x + 12 has two negative roots, and x² − 7x + 12 has two positive roots.
• If c is negative, the roots have opposite signs, one positive and one negative.

This matters because when one variable has both roots positive and the other has both negative, the comparison is settled instantly with no arithmetic: every value of one exceeds every value of the other. Reach for this before you compute magnitudes.

The routine that survives the overlap trap

1. Find both roots of the x equation and both roots of the y equation.
2. Place all four values on a mental number line.
3. Check whether the entire x range sits above, below, or touching the entire y range.
4. If the ranges do not overlap at all, you have a strict relationship: x > y or x < y.
5. If the largest of one equals the smallest of the other, the relationship includes equality: x ≥ y or x ≤ y.
6. If the ranges overlap, so that some x exceed some y while other x fall below other y, no relationship can be established.

The discipline is step three: compare ranges, not single representative values.

Working the equality and no-relationship cases

Equality edges. Suppose x resolves to 2 or 3 and y resolves to 1 or 2. The smallest x is 2 and the largest y is 2, so x is always at least as large as y, with equality possible. The answer is x ≥ y, and marking a strict x > y here is wrong because of the shared value 2. Watch the boundaries specifically.

Overlap. Suppose x is 2 or 5 and y is 3 or 4. Now x = 2 is below both y values, while x = 5 is above both, so neither x > y nor x < y holds across all pairs. The answer is that no relationship can be established. Overlap is not a failure of method; it is a legitimate and frequent answer, and recognising it quickly is worth as much as solving the clean cases.

Where the marks leak

Comparing one root to one root. The overlap case exists precisely to punish this. Always check all four pairings, or equivalently the two ranges.

Missing the equality boundary. When the largest of one set equals the smallest of the other, the answer carries “or equal to”. A strict inequality there is marked wrong.

A sign slip in factorising. Because the comparison depends entirely on the roots, one wrong sign flips the whole answer. Use the sign shortcut as a check: if you factorised x² + 7x + 12 and got positive roots, you have made a sign error, because a positive c with a positive b must give negative roots.

Solving for values you do not need. You need the roots only to compare them. Do not carry decimals to extra places or rationalise anything; magnitudes to enough precision to order them are all the question wants.

Practice that transfers

The way to make the comparison reliable is to do it on paper every time, writing all four values on a line and marking the two ranges, until range-comparison replaces the instinct to compare one number to one number. Keep the factorising fast by drilling calculation speed, since a slow split turns a thirty-second question into a two-minute one.

Work through the quadratic equation practice set, which is built around the overlap and equality cases that decide this topic rather than the clean ones that do not.