Quadratic Equations Practice

Practice recognizing, solving, and interpreting quadratic equations. Quadratic equations include an x² term and often create U-shaped graphs called parabolas.

Skill 1: Recognizing Quadratic Equations

A quadratic equation has a variable raised to the second power. The standard form is ax² + bx + c = 0, where a, b, and c are numbers and a is not zero.

Worked Example

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

Step 1: Look for the highest exponent on x.

The highest exponent is 2.

Step 2: Since the equation contains x², it is quadratic.

Answer: Yes, x² + 3x + 2 = 0 is a quadratic equation.

Try These

  1. Is x² + 5x + 6 = 0 quadratic?
  2. Is 3x + 7 = 0 quadratic?
  3. Is 2x² - 4 = 0 quadratic?
  4. Is x³ + x = 0 quadratic?
  5. Is -x² + 2x - 1 = 0 quadratic?

Answer Key

  1. Yes
  2. No. It is linear.
  3. Yes
  4. No. The highest exponent is 3.
  5. Yes

Mastery Check

You are ready to move on when you can recognize that a quadratic equation contains an x² term and has highest exponent 2.

Skill 2: Solving Quadratics by Factoring

Factoring means rewriting a quadratic as a product of two simpler expressions. If the product equals zero, then at least one factor must equal zero. This is called the zero product property.

Worked Example

Problem: Solve x² + 5x + 6 = 0

Step 1: Factor the quadratic.

x² + 5x + 6 = (x + 2)(x + 3)

Step 2: Set each factor equal to zero.

x + 2 = 0 or x + 3 = 0

Step 3: Solve each equation.

x = -2 or x = -3

Answer: x = -2 and x = -3

Try These

  1. x² + 3x + 2 = 0
  2. x² + 7x + 12 = 0
  3. x² - 5x + 6 = 0
  4. x² - 9 = 0
  5. x² + 6x + 8 = 0

Answer Key

  1. x = -1 and x = -2
  2. x = -3 and x = -4
  3. x = 2 and x = 3
  4. x = -3 and x = 3
  5. x = -2 and x = -4

Mastery Check

You are ready to move on when you can factor a quadratic, set each factor equal to zero, and solve for the possible x-values.

Skill 3: Using the Square Root Method

The square root method is useful when the quadratic equation has an x² term but no x term. To solve, isolate x² first. Then take the square root of both sides. Remember to include both the positive and negative solutions.

Worked Example

Problem: Solve x² = 25

Step 1: Take the square root of both sides.

x = ±√25

Step 2: Simplify the square root.

x = ±5

Answer: x = 5 and x = -5

Try These

  1. x² = 16
  2. x² = 49
  3. x² = 81
  4. x² = 36
  5. x² = 100

Answer Key

  1. x = 4 and x = -4
  2. x = 7 and x = -7
  3. x = 9 and x = -9
  4. x = 6 and x = -6
  5. x = 10 and x = -10

Mastery Check

You are ready to move on when you remember that solving x² = number gives two solutions: one positive and one negative.

Skill 4: Understanding the Quadratic Formula

The quadratic formula can solve any quadratic equation written in standard form: ax² + bx + c = 0. In the formula, a, b, and c come directly from the quadratic equation.

Worked Example

Problem: Identify a, b, and c in x² + 5x + 6 = 0.

Step 1: Compare the equation to standard form.

ax² + bx + c = 0

Step 2: Match each term.

x² means 1x², so a = 1.

5x means b = 5.

6 means c = 6.

Answer: a = 1, b = 5, and c = 6.

Try These

  1. Identify a, b, and c: x² + 3x + 2 = 0
  2. Identify a, b, and c: 2x² - 4x + 1 = 0
  3. Identify a, b, and c: -x² + 6x - 8 = 0
  4. Identify a, b, and c: 5x² + 10 = 0
  5. Identify a, b, and c: 3x² - x - 7 = 0

Answer Key

  1. a = 1, b = 3, c = 2
  2. a = 2, b = -4, c = 1
  3. a = -1, b = 6, c = -8
  4. a = 5, b = 0, c = 10
  5. a = 3, b = -1, c = -7

Mastery Check

You are ready to move on when you can correctly identify a, b, and c from any quadratic equation in standard form.

Skill 5: Interpreting Quadratic Solutions

The solutions of a quadratic equation are also called roots, zeros, or x-intercepts. They tell where the graph crosses the x-axis. A quadratic can have two solutions, one solution, or no real solutions.

Worked Example

Problem: What do the solutions x = -2 and x = -3 mean for the graph?

Step 1: Quadratic solutions are x-values.

Step 2: These x-values show where the graph crosses the x-axis.

Step 3: At an x-intercept, y = 0.

Answer: The graph crosses the x-axis at x = -2 and x = -3.

Try These

  1. If the solutions are x = 1 and x = 4, where does the graph cross the x-axis?
  2. If the solutions are x = -5 and x = 2, where does the graph cross the x-axis?
  3. What is another name for quadratic solutions?
  4. At an x-intercept, what is the value of y?
  5. If a quadratic has one solution, what does that mean about the graph?

Answer Key

  1. The graph crosses the x-axis at x = 1 and x = 4.
  2. The graph crosses the x-axis at x = -5 and x = 2.
  3. Roots, zeros, or x-intercepts.
  4. y = 0
  5. The graph touches the x-axis at one point.

Mastery Check

You are ready to move on when you can explain that quadratic solutions show where the graph crosses or touches the x-axis.

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