Quadratic Equations — Practice

Question 1

Solve 2x² − 7x + 3 = 0 by factorisation.

Accepted Answer

Step 1 — Identify: a = 2, b = −7, c = 3. Product ac = 2 × 3 = 6.

Step 2 — Find two numbers whose product = 6 and sum = −7.
  −6 and −1: product = 6, sum = −7. ✓

Step 3 — Split the middle term.
  2x² − 6x − x + 3 = 0.

Step 4 — Factor by grouping.
  2x(x − 3) − 1(x − 3) = 0.
  (2x − 1)(x − 3) = 0.

Step 5 — Solve.
  x = 1/2 or x = 3.

✦ Answer: x = 1/2 or x = 3.

Question 2

Find the discriminant of 3x² − 5x + 2 = 0 and state the nature of roots.

Accepted Answer

Step 1 — Identify a = 3, b = −5, c = 2.

Step 2 — Compute discriminant.
  Δ = b² − 4ac = (−5)² − 4(3)(2) = 25 − 24 = 1.

Step 3 — Interpret.
  Δ = 1 > 0 and is a perfect square.

✦ Answer: Roots are two distinct REAL and RATIONAL numbers.

Question 3

Solve 2x² + 3x − 5 = 0 using the quadratic formula.

Accepted Answer

Step 1 — Identify a = 2, b = 3, c = −5.

Step 2 — Compute discriminant.
  Δ = b² − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49.

Step 3 — Apply formula.
  x = [−b ± √Δ] / 2a = [−3 ± √49] / 4 = [−3 ± 7] / 4.

Step 4 — Both roots.
  x₁ = (−3 + 7)/4 = 4/4 = 1.
  x₂ = (−3 − 7)/4 = −10/4 = −5/2.

✦ Answer: x = 1 or x = −5/2.

Question 4

Find the value of k for which the equation (k−4)x² + 2(k−4)x + 4 = 0 has equal roots.

Accepted Answer

Step 1 — For equal roots: Δ = 0.
  Here a = (k−4), b = 2(k−4), c = 4.

Step 2 — Apply Δ = 0.
  b² − 4ac = 0.
  [2(k−4)]² − 4(k−4)(4) = 0.
  4(k−4)² − 16(k−4) = 0.

Step 3 — Factor.
  4(k−4)[(k−4) − 4] = 0.
  4(k−4)(k−8) = 0.
  k = 4 or k = 8.

Step 4 — Reject k = 4.
  When k = 4: a = k−4 = 0 → no longer a quadratic equation. REJECT k = 4.

✦ Answer: k = 8.

Verification (k = 8): 4x² + 8x + 4 = 4(x+1)² = 0 → x = −1 (equal root). ✓

Question 5

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, the journey would have taken 1 hour less. Find the speed of the train.

Accepted Answer

Step 1 — Let speed = x km/h.
  Time at speed x = 360/x hours.
  Time at speed (x+5) = 360/(x+5) hours.

Step 2 — Form equation from the time difference.
  360/x − 360/(x+5) = 1.

Step 3 — Simplify.
  360(x+5) − 360x = x(x+5).
  360x + 1800 − 360x = x² + 5x.
  x² + 5x − 1800 = 0.

Step 4 — Factorise.
  Product = −1800, Sum = 5. Factors: 45 and −40.
  (x + 45)(x − 40) = 0.
  x = −45 or x = 40.

Step 5 — Reject negative root.
  Speed cannot be negative. Reject x = −45.

✦ Answer: Speed of the train = 40 km/h.

Verification: Time at 40 km/h = 9 h. Time at 45 km/h = 8 h. Difference = 1 h. ✓

Question 6

Two pipes together can fill a cistern in 3 hours. The larger pipe alone takes 3 hours less than the smaller pipe. Find the time each pipe takes to fill the cistern alone.

Accepted Answer

Step 1 — Let smaller pipe take x hours alone.
  Larger pipe takes (x − 3) hours alone.

Step 2 — Work equation: sum of rates = combined rate.
  1/x + 1/(x−3) = 1/3.

Step 3 — Simplify.
  (x−3 + x) / [x(x−3)] = 1/3.
  3(2x − 3) = x(x − 3).
  6x − 9 = x² − 3x.
  x² − 9x + 9 = 0.

Step 4 — Quadratic formula.
  Δ = 81 − 36 = 45. √45 = 3√5.
  x = (9 ± 3√5)/2.
  x₁ = (9 + 3√5)/2 ≈ (9 + 6.71)/2 ≈ 7.86 h.
  x₂ = (9 − 3√5)/2 ≈ (9 − 6.71)/2 ≈ 1.15 h.

Step 5 — Check validity.
  If x ≈ 1.15, then larger pipe = x − 3 ≈ −1.85 h (negative — REJECT).
  Take x = (9 + 3√5)/2 hours (smaller pipe), larger = (3 + 3√5)/2 hours.

✦ Answer: Smaller pipe = (9+3√5)/2 ≈ 7.86 h, Larger pipe = (3+3√5)/2 ≈ 4.86 h.

Question 7

Solve x² + 6x − 7 = 0 by completing the square.

Accepted Answer

Step 1 — Move constant to RHS.
  x² + 6x = 7.

Step 2 — Add (half of coefficient of x)² to both sides.
  Half of 6 = 3. 3² = 9.
  x² + 6x + 9 = 7 + 9 = 16.

Step 3 — Write LHS as perfect square.
  (x + 3)² = 16.

Step 4 — Take square root.
  x + 3 = ±4.
  x = −3 + 4 = 1 or x = −3 − 4 = −7.

✦ Answer: x = 1 or x = −7.

Practice — Quadratic Equations

Sign in to save attempts to your dashboard and track your accuracy over time

Find a verified tutor

Join a live course

Browse coaching centres

About

Categories

Support

Documentation

Get Study Tips & Updates

Download Our App

Download for iOS

Download for Android

Popular Cities