Quadratic Equations for Bank Exams: Method and 10 MCQs

Bank papers give two quadratic equations, one in x and one in y, and ask how x and y compare. Each question takes about a minute once the method is automatic.

Method

1. Write each equation in the form ax² + bx + c = 0
2. Find two numbers that multiply to ac and add to b, then split the middle term
3. Read off the roots from the factors
4. Compare all pairs. If every x is smaller than every y, then x < y. If some pairs go each way, the relation cannot be established

Worked example

I: x² − 7x + 12 = 0 gives (x − 3)(x − 4) = 0, so x = 3 or 4. II: y² − 9y + 20 = 0 gives (y − 4)(y − 5) = 0, so y = 4 or 5. Every x is at most every y, so x ≤ y.

Three ways to solve a quadratic

• Factorisation: split the middle term. For ax² + bx + c, find two numbers whose product is a × c and whose sum is b. For x² − 5x + 6 = 0, the numbers are −2 and −3, so (x − 2)(x − 3) = 0 and x = 2 or 3
• The formula: x = (−b ± √(b² − 4ac)) ÷ 2a. Use it when factorising is not obvious
• Completing the square: useful for deriving the formula and for some maximum and minimum problems

Worked examples

• 2x² − 7x + 3 = 0: split −7x as −6x − x to get 2x(x − 3) − 1(x − 3) = 0, so (2x − 1)(x − 3) = 0 and x = 1/2 or 3
• x² + x − 12 = 0: the numbers 4 and −3 multiply to −12 and add to 1, so (x + 4)(x − 3) = 0 and x = −4 or 3
• x² = 16: x = 4 or −4

Sum and product of roots

For ax² + bx + c = 0, the sum of the roots is −b ÷ a and the product is c ÷ a. For x² − 7x + 12 = 0, the sum is 7 and the product is 12, so the roots are 3 and 4. This lets you check an answer in seconds and build an equation from given roots.

The discriminant

D = b² − 4ac. If D > 0, there are two distinct real roots. If D = 0, the roots are real and equal. If D < 0, there are no real roots. For x² + 2x + 5 = 0, D = 4 − 20 = −16, so there are no real roots.

Sign shortcuts

• If c is positive and b is negative, both roots are positive
• If c is positive and b is positive, both roots are negative
• If c is negative, one root is positive and one is negative, and the larger root takes the sign of −b

The bank exam format: compare x and y

A bank paper usually gives two equations, one in x and one in y, and asks for the relation between x and y. Solve each, write the roots, and compare every x with every y. Example: I. x² − 9x + 20 = 0 gives x = 4 or 5. II. y² − 11y + 30 = 0 gives y = 5 or 6. Every x is less than or equal to every y, so x ≤ y. If the roots overlap in both directions, the answer is that no relation can be established.

Common mistakes

• Getting the signs of the roots wrong after factorising
• Forgetting the negative root when a square equals a positive number
• Comparing only one pair of roots
• Spending too long on factorising when the formula would be faster

Practice routine

Solve ten equations a day for a week, mixing the three methods. Time yourself at 60 seconds per question, and once you can do that, try x and y comparison sets at 90 seconds each.

Practise

Try the 10 MCQs below, then continue with Quadratic Equations practice in the J&K Bank question bank.