Quadratic Functions and Complex Numbers — High School Algebra 2 practice questions

Solve x² − 4x + 1 = 0.

1. x = 2 ± √3
2. x = 4 ± 2√3
3. x = 2 ± 2√3
4. x = −2 ± √3

Answer: A — x = 2 ± √3

A) Correct — x = (4 ± √12)/2 = (4 ± 2√3)/2. B) Forgets to divide by 2a. C) Divides only the 4 by 2. D) Uses b instead of −b.

What is the axis of symmetry of y = 2x² − 8x + 1?

1. x = −2
2. x = 4
3. x = 8
4. x = 2

Answer: D — x = 2

D) Correct — x = −b/(2a) = 8/4. A) Drops the negative in −b/(2a). B) Divides by a instead of 2a. C) Uses −b alone.

Solve x² + 4 = 0.

1. x = ±2
2. x = ±2i
3. x = −4
4. No solution

Answer: B — x = ±2i

B) Correct — x² = −4, so x = ±√(−4) = ±2i. A) Ignores the negative under the root. C) Stops at x² = −4. D) Has no real solutions, but has complex ones.

Simplify (3 + 2i) + (5 − 4i).

1. 8 − 2i
2. 8 + 6i
3. 8 + 2i
4. 15 − 8i

Answer: A — 8 − 2i

A) Correct — Add real parts and imaginary parts separately. B) Adds 2i and 4i, ignoring the minus sign. C) Gets the sign of the imaginary part wrong. D) Multiplies instead of adding.

Simplify (3 + 2i)(1 − 4i).

1. −5 − 10i
2. 11 − 12i
3. 11 − 10i
4. 11 + 10i

Answer: C — 11 − 10i

C) Correct — 3 − 12i + 2i − 8i² = 3 − 10i + 8. A) Treats i² as +1. B) Leaves out the 2i term. D) Gets the sign of the imaginary part wrong.

Solve x² − 6x + 13 = 0.

1. x = 3 ± 2i
2. x = 3 ± 4i
3. x = −3 ± 2i
4. x = 6 ± 2i

Answer: A — x = 3 ± 2i

A) Correct — x = (6 ± √(−16))/2 = (6 ± 4i)/2. B) Divides only the 6 by 2. C) Uses b instead of −b. D) Divides only the 4i by 2.

What is the vertex of f(x) = x² − 6x + 5?

1. (−3, −4)
2. (3, −4)
3. (3, 5)
4. (6, 5)

Answer: B — (3, −4)

B) Correct — x = −b/(2a) = 3, and f(3) = 9 − 18 + 5 = −4. A) Drops the negative in −b/(2a). C) Uses c as the y-value. D) Uses −b and c directly.

What is i²⁷?

1. i
2. −1
3. −i
4. 1

Answer: C — −i

C) Correct — Powers of i repeat every 4; 27 has remainder 3, and i³ = −i. A) Uses remainder 1. B) Uses remainder 2. D) Assumes 27 is a multiple of 4.

What number completes the square for x² + 8x?

1. 8
2. 4
3. 16
4. 64

Answer: C — 16

C) Correct — Add (b/2)² = 4². A) Uses b without halving and squaring. B) Halves b but doesn't square it. D) Squares b without halving it.

What is the complex conjugate of 3 + 2i?

1. −3 − 2i
2. −3 + 2i
3. 2 + 3i
4. 3 − 2i

Answer: D — 3 − 2i

D) Correct — Change the sign of the imaginary part. A) Changes both signs. B) Changes the real part instead. C) Swaps the parts.

What's i³ ?

−i.

Worked Example: Write the equation of the circle with center (−2, 5) and radius 4.

Step 1: Recall the standard form of a circle. (x − h)² + (y − k)² = r² where (h, k) is the center and r is the radius. Step 2: Substitute h = −2, k = 5, r = 4. (x − (−2))² + (y − 5)² = 4² Step 3: Simplify the signs. (x + 2)² + (y − 5)² = 16 Step 4: Verify by checking a point on the circle. The point (2, 5) should be 4 units from (−2, 5). Distance = √((2 − (−2))² + (5 − 5)²) = √16 = 4 ✓ Final answer: (x + 2)² + (y − 5)² = 16.

What does the graph of y = ax² + bx + c look like?

A parabola. Opens up if a > 0, down if a < 0.

What is the 'sum and product of roots' for ax² + bx + c = 0?

Sum = −b/a, product = c/a.