Quadratic Equations — Practice

Question 1

Is x² + 2x − 3 = 0 a quadratic equation?

Accepted Answer

✦ Answer: YES. a=1, b=2, c=−3, a ≠ 0. Standard form: ax² + bx + c = 0.

Question 2

Find discriminant of 2x² − 3x + 1 = 0.

Accepted Answer

✦ Answer: D = b² − 4ac = (−3)² − 4(2)(1) = 9 − 8 = 1 > 0. So two distinct real roots.

Question 3

Solve by quadratic formula: 2x² − 7x + 3 = 0.

Accepted Answer

Step 1 — Identify coefficients.
  a = 2, b = −7, c = 3

Step 2 — Compute discriminant.
  D = b² − 4ac = 49 − 24 = 25 > 0 (distinct real roots)

Step 3 — Apply formula.
  x = (−(−7) ± √25) / (2 × 2)
  x = (7 ± 5) / 4

Step 4 — Two solutions.
  x = (7 + 5)/4 = 12/4 = 3
  x = (7 − 5)/4 = 2/4 = 1/2

Step 5 — Verify.
  For x = 3: 2(9) − 21 + 3 = 18 − 21 + 3 = 0 ✓
  For x = 1/2: 2(1/4) − 7/2 + 3 = 1/2 − 7/2 + 3 = −3 + 3 = 0 ✓

✦ Answer: x = 3 or x = 1/2.

Question 4

The product of two consecutive positive integers is 240. Find the integers.

Accepted Answer

Step 1 — Set up variables.
  Let smaller integer = x. Then next consecutive integer = x + 1.

Step 2 — Form equation.
  x(x + 1) = 240
  x² + x − 240 = 0

Step 3 — Solve by quadratic formula.
  a = 1, b = 1, c = −240
  D = 1 + 960 = 961
  √961 = 31
  x = (−1 ± 31) / 2

Step 4 — Find roots.
  x = 30/2 = 15 (positive)
  x = −32/2 = −16 (rejected, must be positive)

Step 5 — Find integers.
  x = 15, x + 1 = 16

Step 6 — Verify.
  15 × 16 = 240 ✓

✦ Answer: The two consecutive positive integers are 15 and 16.

Practice — Quadratic Equations