Solving Quadratic Equations Flashcards

What are the solutions to x² - 5x + 6 = 0?

Answer: x = 2 and x = 3

Factoring gives (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Using the quadratic formula, what is the discriminant of 3x² - 4x + 2 = 0?

Answer: −8

Discriminant = b² − 4ac = 16 − 24 = −8.

Which of the following equations has exactly one repeated root?

Answer: x² − 6x + 9 = 0

x² − 6x + 9 = (x − 3)² = 0 gives the repeated root x = 3.

Solve 2x² + 7x + 3 = 0 by factoring.

Answer: x = −3 and x = −½

Factoring gives (2x + 1)(x + 3) = 0, so x = −½ or x = −3.

What is the vertex form of x² − 4x + 7?

Answer: (x − 2)² + 3

Completing the square: x² − 4x + 4 − 4 + 7 = (x − 2)² + 3.

How many real solutions does x² + 4 = 0 have?

Answer: None

x² = −4 has no real solutions because a square cannot be negative.

A square has side length (x − 2). Its area equals 3x + 2. What positive value of x satisfies this?

Answer: 8

Expanding gives x² − 4x + 4 = 3x + 2, leading to x² − 7x + 2 = 0; the positive root is approximately x = 8.