Quadratic Equations — Practice

Question 1

Is (x − 2)² = x² − 4 a quadratic equation? Justify.

Accepted Answer

Step 1 — Expand LHS: x² − 4x + 4 = x² − 4.

Step 2 — Simplify: −4x + 4 = −4 ⇒ −4x + 8 = 0 (the x² terms cancel).

Step 3 — No x² term remains, so it is linear, not quadratic.

✦ Answer: No — it reduces to a linear equation.

Question 2

Solve x² − 5x + 6 = 0 by factorisation.

Accepted Answer

Step 1 — Two numbers with product 6, sum −5: −2 and −3.

Step 2 — (x − 2)(x − 3) = 0.

✦ Answer: x = 2 or x = 3.

Question 3

Find the discriminant of 2x² − 4x + 3 = 0 and state the nature of its roots.

Accepted Answer

Step 1 — D = b² − 4ac = (−4)² − 4(2)(3) = 16 − 24 = −8.

Step 2 — D < 0 ⇒ no real roots.

✦ Answer: D = −8; no real roots.

Question 4

Solve 6x² − x − 2 = 0 by splitting the middle term.

Accepted Answer

Step 1 — Product ac = 6×(−2) = −12, sum = −1 → numbers −4 and 3.

Step 2 — 6x² − 4x + 3x − 2 = 0 ⇒ 2x(3x − 2) + 1(3x − 2) = 0 ⇒ (3x − 2)(2x + 1) = 0.

Step 3 — x = 2/3 or x = −1/2.

✦ Answer: x = 2/3 or x = −1/2.

Question 5

Solve 2x² − 7x + 3 = 0 using the quadratic formula.

Accepted Answer

Step 1 — a = 2, b = −7, c = 3; D = 49 − 24 = 25.

Step 2 — x = [7 ± √25]/4 = (7 ± 5)/4.

Step 3 — x = 12/4 = 3 or x = 2/4 = 1/2.

✦ Answer: x = 3 or x = 1/2.

Question 6

Solve x² + 4x − 5 = 0 by completing the square.

Accepted Answer

Step 1 — x² + 4x = 5.

Step 2 — Add (4/2)² = 4 to both sides: x² + 4x + 4 = 9 ⇒ (x + 2)² = 9.

Step 3 — x + 2 = ±3 ⇒ x = 1 or x = −5.

✦ Answer: x = 1 or x = −5.

Question 7

Find the value(s) of k for which the equation kx² − 2kx + 6 = 0 has equal roots (k ≠ 0).

Accepted Answer

Step 1 — Equal roots ⇒ D = 0. Here a = k, b = −2k, c = 6.

Step 2 — D = (−2k)² − 4(k)(6) = 4k² − 24k = 0.

Step 3 — 4k(k − 6) = 0 ⇒ k = 0 or k = 6. Since k ≠ 0, k = 6.

✦ Answer: k = 6.

Question 8

The length of a rectangular plot is one metre more than twice its breadth, and its area is 528 m². Find its dimensions.

Accepted Answer

Step 1 — Let breadth = x m; length = (2x + 1) m.

Step 2 — Area: x(2x + 1) = 528 ⇒ 2x² + x − 528 = 0.

Step 3 — D = 1 + 4(2)(528) = 1 + 4224 = 4225; √D = 65.

Step 4 — x = (−1 ± 65)/4 ⇒ x = 16 or x = −16.5. Reject the negative breadth.

Step 5 — Breadth = 16 m, length = 2(16) + 1 = 33 m.

✦ Answer: 16 m × 33 m.

Question 9

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

Accepted Answer

Step 1 — Let speed = x km/h. Time = 360/x.

Step 2 — At (x + 5) km/h: time = 360/(x + 5), which is 1 hour less.

Step 3 — 360/x − 360/(x + 5) = 1 ⇒ 360(x + 5 − x) = x(x + 5) ⇒ 1800 = x² + 5x.

Step 4 — x² + 5x − 1800 = 0; D = 25 + 7200 = 7225; √D = 85.

Step 5 — x = (−5 ± 85)/2 ⇒ x = 40 or x = −45. Reject negative speed.

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

Question 10

The product of two consecutive positive integers is 306. Form a quadratic equation and find the integers.

Accepted Answer

Step 1 — Let the smaller integer be x; the next is (x + 1).

Step 2 — Product: x(x + 1) = 306 ⇒ x² + x − 306 = 0.

Step 3 — D = 1² − 4(1)(−306) = 1 + 1224 = 1225; √D = 35.

Step 4 — x = (−1 ± 35)/2 ⇒ x = 17 or x = −18. Reject the negative (integers must be positive).

Step 5 — So x = 17 and the next integer is 18.

✦ Answer: x² + x − 306 = 0; the integers are 17 and 18.

Practice — Quadratic Equations

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