Practice recognizing, solving, and interpreting quadratic equations. Quadratic equations include an x² term and often create U-shaped graphs called parabolas.
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.
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.
You are ready to move on when you can recognize that a quadratic equation contains an x² term and has highest exponent 2.
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.
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
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.
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.
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
You are ready to move on when you remember that solving x² = number gives two solutions: one positive and one negative.
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.
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.
You are ready to move on when you can correctly identify a, b, and c from any quadratic equation in standard form.
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.
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.
You are ready to move on when you can explain that quadratic solutions show where the graph crosses or touches the x-axis.