InterviewSam
Log inStart free

Linear & Quadratic Equations

Solving equations, roots and comparing quantities.

45 solved questions · 15 easy · 20 medium · 10 hard

Take a 10-question quiz on Linear & Quadratic Equations, no login

Shortcuts & formulas

  • Quadratic ax² + bx + c = 0: sum of roots = -b/a, product of roots = c/a.
  • Roots by formula: x = [-b ± √(b² - 4ac)] / 2a. Discriminant D = b² - 4ac: D > 0 two real roots, D = 0 equal roots, D < 0 no real roots.
  • Factorising x² + (p+q)x + pq = (x+p)(x+q): find two numbers with the given product and sum.
  • α² + β² = (α+β)² - 2αβ and α³ + β³ = (α+β)³ - 3αβ(α+β).
  • For comparison questions solve both equations, list all roots, and compare every x with every y; if the order changes, the relation cannot be established.
  • x + 1/x = k gives x² + 1/x² = k² - 2.
  • Two simultaneous equations: eliminate one variable by making coefficients equal, then add or subtract.
  • Consecutive integers: n, n+1, n+2; their sum is 3 times the middle one.

Solved examples

Try each one first, then open the answer and the step-by-step solution.

Easy questions

Q1.

Solve for x: 3x + 7 = 28.

  1. A7
  2. B9
  3. C21
  4. D5
Show answer & solution

Answer: A. 7

Solution

3x = 28 - 7 = 21 x = 21/3 = 7 Answer: 7

Q2.

If 5x - 12 = 3x + 8, find x.

  1. A4
  2. B10
  3. C2
  4. D20
Show answer & solution

Answer: B. 10

Solution

5x - 3x = 8 + 12 2x = 20 x = 10 Answer: 10

Q3.

The sum of two numbers is 45 and their difference is 9. Find the larger number.

  1. A36
  2. B18
  3. C27
  4. D24
Show answer & solution

Answer: C. 27

Solution

Larger = (45 + 9)/2 = 27 Smaller = (45 - 9)/2 = 18 Answer: 27

Shortcut: Larger = (sum + difference)/2.

Q4.

Three times a number increased by 8 equals 50. What is the number?

  1. A16
  2. B19
  3. C12
  4. D14
Show answer & solution

Answer: D. 14

Solution

3x + 8 = 50 3x = 42 x = 14 Answer: 14

Q5.

What is the sum of the roots of x² - 9x + 20 = 0?

  1. A9
  2. B20
  3. C29
  4. D11
Show answer & solution

Answer: A. 9

Solution

Sum of roots = -b/a = -(-9)/1 = 9 (Roots are 4 and 5, 4 + 5 = 9) Answer: 9

Q6.

What is the product of the roots of x² - 7x + 12 = 0?

  1. A7
  2. B12
  3. C24
  4. D6
Show answer & solution

Answer: B. 12

Solution

Product of roots = c/a = 12/1 = 12 (Roots 3 and 4) Answer: 12

Q7.

Find the roots of x² - 5x + 6 = 0.

  1. A1 and 6
  2. B3 and 4
  3. C2 and 3
  4. D2 and 4
Show answer & solution

Answer: C. 2 and 3

Solution

x² - 5x + 6 = (x - 2)(x - 3) = 0 x = 2 or x = 3 Answer: 2 and 3

Q8.

Find the larger root of x² + 2x - 15 = 0.

  1. A5
  2. B15
  3. C2
  4. D3
Show answer & solution

Answer: D. 3

Solution

x² + 2x - 15 = (x + 5)(x - 3) = 0 x = -5 or 3 Larger root = 3 Answer: 3

Q9.

Find the sum of three consecutive integers if the smallest is 17.

  1. A54
  2. B51
  3. C52
  4. D57
Show answer & solution

Answer: A. 54

Solution

Numbers: 17, 18, 19 Sum = 54 Answer: 54

Q10.

A father is 4 times as old as his son. The sum of their ages is 60 years. Find the son's age.

  1. A15
  2. B12
  3. C48
  4. D10
Show answer & solution

Answer: B. 12

Solution

Let son = x, father = 4x 5x = 60 x = 12 Answer: 12 years

Q11.

If 4 pens and 3 pencils cost ₹50 and each pen costs ₹8, find the cost of one pencil.

  1. A5
  2. B8
  3. C6
  4. D7
Show answer & solution

Answer: C. 6

Solution

4 pens = 4 x 8 = ₹32 3 pencils = 50 - 32 = ₹18 One pencil = ₹6 Answer: ₹6

Q12.

Find the discriminant of 2x² - 5x + 3 = 0.

  1. A49
  2. B25
  3. C7
  4. D1
Show answer & solution

Answer: D. 1

Solution

D = b² - 4ac = 25 - 4(2)(3) = 25 - 24 = 1 Answer: 1

Q13.

Solve: x/3 + x/6 = 9.

  1. A18
  2. B27
  3. C15
  4. D9
Show answer & solution

Answer: A. 18

Solution

LCM 6: (2x + x)/6 = 9 3x/6 = 9 so x/2 = 9 x = 18 Answer: 18

Q14.

If 2x + 3y = 12 and x = 3, find y.

  1. A3
  2. B2
  3. C6
  4. D1
Show answer & solution

Answer: B. 2

Solution

2(3) + 3y = 12 3y = 6 y = 2 Answer: 2

Q15.

Which statement is true for x² + 4x + 5 = 0?

  1. ATwo equal real roots
  2. BTwo distinct real roots because D = 36
  3. CNo real roots because D = -4
  4. DNo real roots because D = 16
Show answer & solution

Answer: C. No real roots because D = -4

Solution

D = 16 - 4(1)(5) = 16 - 20 = -4 < 0 So there are no real roots. Answer: No real roots because D = -4

Medium questions

Q16.

Solve the two equations and compare x and y. I. x² - 7x + 12 = 0 II. y² - 3y + 2 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: A. x is greater than y

Solution

I. x² - 7x + 12 = 0 gives x = 3, 4 II. y² - 3y + 2 = 0 gives y = 1, 2 Every value of x is greater than every value of y. Answer: x is greater than y

Shortcut: List the roots on a number line; compare the smallest and largest of each side.

Q17.

Solve the two equations and compare x and y. I. x² + 7x + 10 = 0 II. y² - 4y + 3 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: B. x is less than y

Solution

I. x² + 7x + 10 = 0 gives x = -5, -2 II. y² - 4y + 3 = 0 gives y = 1, 3 Every value of x is smaller than every value of y. Answer: x is less than y

Shortcut: List the roots on a number line; compare the smallest and largest of each side.

Q18.

Solve the two equations and compare x and y. I. x² - 9 = 0 II. y² - y - 2 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: D. Relation cannot be established

Solution

I. x² - 9 = 0 gives x = -3, 3 II. y² - y - 2 = 0 gives y = -1, 2 x is greater than y for some pairs and smaller for others, so no fixed relation can be established. Answer: Relation cannot be established

Shortcut: List the roots on a number line; compare the smallest and largest of each side.

Q19.

Solve the two equations and compare x and y. I. x² - 8x + 16 = 0 II. y² - 8y + 16 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: C. x equals y

Solution

I. x² - 8x + 16 = 0 gives x = 4 II. y² - 8y + 16 = 0 gives y = 4 x = y in every case. Answer: x equals y

Shortcut: List the roots on a number line; compare the smallest and largest of each side.

Q20.

Solve the two equations and compare x and y. I. x² - 9x + 14 = 0 II. y² - 4y + 3 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: D. Relation cannot be established

Solution

I. x² - 9x + 14 = 0 gives x = 2, 7 II. y² - 4y + 3 = 0 gives y = 1, 3 x is greater than y for some pairs and smaller for others, so no fixed relation can be established. Answer: Relation cannot be established

Shortcut: List the roots on a number line; compare the smallest and largest of each side.

Q21.

Solve the two equations and compare x and y. I. x² + 7x + 6 = 0 II. y² + 6y + 8 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: D. Relation cannot be established

Solution

I. x² + 7x + 6 = 0 gives x = -6, -1 II. y² + 6y + 8 = 0 gives y = -4, -2 x is greater than y for some pairs and smaller for others, so no fixed relation can be established. Answer: Relation cannot be established

Shortcut: List the roots on a number line; compare the smallest and largest of each side.

Q22.

Solve the two equations and compare x and y. I. 3x - 15 = 0 II. y² - 4y + 3 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: A. x is greater than y

Solution

I. 3x - 15 = 0 gives x = 5 II. y² - 4y + 3 = 0 gives y = 1, 3 Every value of x is greater than every value of y. Answer: x is greater than y

Shortcut: List the roots on a number line; compare the smallest and largest of each side.

Q23.

Solve the two equations and compare x and y. I. x² - 49 = 0 II. y² - 17y + 72 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: B. x is less than y

Solution

I. x² - 49 = 0 gives x = -7, 7 II. y² - 17y + 72 = 0 gives y = 8, 9 Every value of x is smaller than every value of y. Answer: x is less than y

Shortcut: List the roots on a number line; compare the smallest and largest of each side.

Q24.

If 2x + 3y = 17 and x - y = 1, find x.

  1. A3
  2. B5
  3. C2
  4. D4
Show answer & solution

Answer: D. 4

Solution

From x - y = 1: y = x - 1 2x + 3(x - 1) = 17 5x = 20 x = 4 (and y = 3) Answer: 4

Q25.

If 3x + 2y = 26 and 5x - 2y = 6, find x + y.

  1. A11
  2. B9
  3. C10
  4. D12
Show answer & solution

Answer: A. 11

Solution

Add the equations: 8x = 32 so x = 4 3(4) + 2y = 26 so 2y = 14, y = 7 x + y = 11 Answer: 11

Q26.

The product of two consecutive positive integers is 156. Find their sum.

  1. A24
  2. B25
  3. C26
  4. D23
Show answer & solution

Answer: B. 25

Solution

n(n + 1) = 156 so n² + n - 156 = 0 (n + 13)(n - 12) = 0, n = 12 Numbers 12 and 13, sum = 25 Answer: 25

Shortcut: Find the number whose square is just below 156: 12 x 13 = 156.

Q27.

The product of two consecutive even positive integers is 168. Find their sum.

  1. A25
  2. B28
  3. C26
  4. D24
Show answer & solution

Answer: C. 26

Solution

n(n + 2) = 168 so n² + 2n - 168 = 0 (n + 14)(n - 12) = 0, n = 12 Numbers 12 and 14, sum = 26 Answer: 26

Q28.

The length of a rectangle is 3 m more than its breadth and its area is 70 m². Find its perimeter (in m).

  1. A17
  2. B32
  3. C20
  4. D34
Show answer & solution

Answer: D. 34

Solution

Let breadth = b, then b(b + 3) = 70 b² + 3b - 70 = 0, (b + 10)(b - 7) = 0, b = 7 Length = 10 Perimeter = 2(7 + 10) = 34 m Answer: 34 m

Q29.

A rectangle has perimeter 36 cm and area 77 cm². Find its longer side (in cm).

  1. A11
  2. B9
  3. C7
  4. D14
Show answer & solution

Answer: A. 11

Solution

l + b = 18 and lb = 77 l and b are roots of t² - 18t + 77 = 0 (t - 7)(t - 11) = 0 Longer side = 11 cm Answer: 11 cm

Q30.

For what positive value of k does x² + kx + 25 = 0 have equal roots?

  1. A5
  2. B10
  3. C25
  4. D20
Show answer & solution

Answer: B. 10

Solution

Equal roots need D = 0 k² - 4(1)(25) = 0 k² = 100, k = 10 (positive) Answer: 10

Q31.

For what value of k does 2x² - 8x + k = 0 have equal roots?

  1. A4
  2. B16
  3. C8
  4. D2
Show answer & solution

Answer: C. 8

Solution

D = 64 - 8k = 0 k = 8 Answer: 8

Q32.

If α and β are the roots of x² - 8x + 10 = 0, find α² + β².

  1. A54
  2. B64
  3. C34
  4. D44
Show answer & solution

Answer: D. 44

Solution

α + β = 8, αβ = 10 α² + β² = (α + β)² - 2αβ = 64 - 20 = 44 Answer: 44

Shortcut: Square the sum, subtract twice the product.

Q33.

If α and β are the roots of x² - 5x + 3 = 0, find 1/α + 1/β.

  1. A5/3
  2. B3/5
  3. C5/2
  4. D15
Show answer & solution

Answer: A. 5/3

Solution

α + β = 5, αβ = 3 1/α + 1/β = (α + β)/αβ = 5/3 Answer: 5/3

Q34.

The present ages of A and B are in the ratio 3 : 5. After 6 years the ratio becomes 2 : 3. Find B's present age (in years).

  1. A18
  2. B30
  3. C24
  4. D36
Show answer & solution

Answer: B. 30

Solution

Let ages be 3k and 5k (3k + 6)/(5k + 6) = 2/3 9k + 18 = 10k + 12 so k = 6 B = 5 x 6 = 30 years Answer: 30

Q35.

If x + y = 10 and xy = 21, find x² + y².

  1. A79
  2. B100
  3. C58
  4. D42
Show answer & solution

Answer: C. 58

Solution

x² + y² = (x + y)² - 2xy = 100 - 42 = 58 Answer: 58

Hard questions

Q36.

Solve the two equations and compare x and y. I. 2x² - 7x + 6 = 0 II. 3y² - 11y + 10 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: D. Relation cannot be established

Solution

I. 2x² - 7x + 6 = 0 gives x = 3/2, 2 II. 3y² - 11y + 10 = 0 gives y = 5/3, 2 x is greater than y for some pairs and smaller for others, so no fixed relation can be established. Answer: Relation cannot be established

Shortcut: With fractional roots, convert to decimals first (e.g. 3/2 = 1.5, 5/3 = 1.67).

Q37.

Solve the two equations and compare x and y. I. 2x² - 9x + 10 = 0 II. 4y² - 8y + 3 = 0

  1. Ax is greater than y
  2. Bx is less than y
  3. Cx equals y
  4. DRelation cannot be established
Show answer & solution

Answer: A. x is greater than y

Solution

I. 2x² - 9x + 10 = 0 gives x = 2, 5/2 II. 4y² - 8y + 3 = 0 gives y = 1/2, 3/2 Every value of x is greater than every value of y. Answer: x is greater than y

Shortcut: With fractional roots, convert to decimals first (e.g. 3/2 = 1.5, 5/3 = 1.67).

Q38.

If x + 1/x = 5, find x² + 1/x².

  1. A25
  2. B23
  3. C27
  4. D21
Show answer & solution

Answer: B. 23

Solution

(x + 1/x)² = x² + 2 + 1/x² 25 = x² + 1/x² + 2 x² + 1/x² = 23 Answer: 23

Q39.

If x + 1/x = 4, find x³ + 1/x³.

  1. A64
  2. B60
  3. C52
  4. D48
Show answer & solution

Answer: C. 52

Solution

x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x) = 64 - 12 = 52 Answer: 52

Q40.

Pipe A alone fills a tank 6 hours faster than pipe B alone. Together they fill it in 4 hours. How long does A alone take (in hours)?

  1. A8
  2. B12
  3. C4
  4. D6
Show answer & solution

Answer: D. 6

Solution

Let A take t hours, B takes t + 6 1/t + 1/(t + 6) = 1/4 4(2t + 6) = t(t + 6) t² - 2t - 24 = 0, (t - 6)(t + 4) = 0, t = 6 Check: 1/6 + 1/12 = 1/4 Answer: 6 hours

Q41.

A train covers 120 km at a uniform speed. Had its speed been 10 km/h more, it would have taken 1 hour less. Find its speed (in km/h).

  1. A30
  2. B20
  3. C40
  4. D25
Show answer & solution

Answer: A. 30

Solution

120/v - 120/(v + 10) = 1 1200 = v(v + 10), v² + 10v - 1200 = 0 (v + 40)(v - 30) = 0, v = 30 Check: 120/30 = 4 h, 120/40 = 3 h Answer: 30 km/h

Q42.

If α and β are the roots of 2x² - 6x + 3 = 0, find α/β + β/α.

  1. A3
  2. B4
  3. C6
  4. D2
Show answer & solution

Answer: B. 4

Solution

α + β = 3, αβ = 3/2 α² + β² = 9 - 3 = 6 α/β + β/α = 6/(3/2) = 4 Answer: 4

Q43.

In x² - 9x + k = 0 one root is twice the other. Find k.

  1. A14
  2. B20
  3. C18
  4. D27
Show answer & solution

Answer: C. 18

Solution

Let roots be a and 2a: 3a = 9 so a = 3 Roots 3 and 6, k = 3 x 6 = 18 Answer: 18

Q44.

A quadratic has roots that are each 2 more than the roots of x² - 5x + 6 = 0. Which equation is it?

  1. Ax² - 9x + 18 = 0
  2. Bx² - 5x + 8 = 0
  3. Cx² - 7x + 12 = 0
  4. Dx² - 9x + 20 = 0
Show answer & solution

Answer: D. x² - 9x + 20 = 0

Solution

Original roots 2, 3; new roots 4, 5 Sum = 9, product = 20 Equation: x² - 9x + 20 = 0 Answer: x² - 9x + 20 = 0

Q45.

A positive number exceeds its reciprocal by 8/3. Find the number.

  1. A3
  2. B2
  3. C4
  4. D1/3
Show answer & solution

Answer: A. 3

Solution

x - 1/x = 8/3 3x² - 3 = 8x, 3x² - 8x - 3 = 0 (3x + 1)(x - 3) = 0, x = 3 or -1/3 Positive number = 3 Check: 3 - 1/3 = 8/3 Answer: 3

Ready to test yourself? 10 fresh Linear & Quadratic Equations questions, timed, with solutions at the end.

Start the quiz

Related topics

Number SystemHCF & LCMSimplification & ApproximationPercentagesProfit, Loss & DiscountSimple & Compound InterestRatio & ProportionAveragesMixtures & AlligationPartnershipTime & WorkPipes & CisternsTime, Speed & DistanceBoats & StreamsProblems on AgesPermutations & CombinationsProbabilityMensurationToday's Daily Quiz · 5 questions

Practise this with Sam

A real mock interview, out loud, with an honest scorecard. Free to start.

Start free