Learning how to solve quadratic equations takes practice. Whether you’re using factoring, completing the square, or the quadratic formula, you need to know the steps and apply them correctly.
Here are the best practice problems with solutions to help you sharpen your skills. Each one includes clear steps so you can learn from the process.
Problem 1: Factoring
Solve:
x² – 7x + 12 = 0
Step 1: Find two numbers that multiply to 12 and add to -7
Those numbers are -3 and -4
Step 2: Factor the equation
(x – 3)(x – 4) = 0
Step 3: Set each factor equal to 0
x – 3 = 0 → x = 3
x – 4 = 0 → x = 4
Solution:
x = 3 or x = 4
Problem 2: Quadratic Formula
Solve:
2x² – 4x – 6 = 0
Step 1: Identify a, b, and c
a = 2, b = -4, c = -6
Step 2: Use the formula
x = (-b ± √(b² – 4ac)) / 2a
Step 3: Plug in the values
x = (4 ± √((-4)² – 4(2)(-6))) / (2 × 2)
x = (4 ± √(16 + 48)) / 4
x = (4 ± √64) / 4
x = (4 ± 8) / 4
Step 4: Solve both options
x = (4 + 8)/4 = 12/4 = 3
x = (4 – 8)/4 = -4/4 = -1
Solution:
x = 3 or x = -1
Problem 3: Completing the Square
Solve:
x² + 6x + 5 = 0
Step 1: Move the constant to the other side
x² + 6x = -5
Step 2: Take half of 6, square it
(6/2)² = 9
Step 3: Add 9 to both sides
x² + 6x + 9 = 4
Step 4: Write the left side as a square
(x + 3)² = 4
Step 5: Take the square root of both sides
x + 3 = ±2
Step 6: Solve both options
x = -3 + 2 = -1
x = -3 – 2 = -5
Solution:
x = -1 or x = -5
Problem 4: Disguised Quadratic
Solve:
(x – 1)(x + 4) = 12
Step 1: Expand the left side
x² + 4x – x – 4 = 12 → x² + 3x – 4 = 12
Step 2: Move 12 to the left side
x² + 3x – 16 = 0
Step 3: Try factoring
Find two numbers that multiply to -16 and add to 3
Not possible, so use the formula
Step 4: a = 1, b = 3, c = -16
x = (-3 ± √(3² – 4(1)(-16))) / 2(1)
x = (-3 ± √(9 + 64)) / 2
x = (-3 ± √73) / 2
Solution:
x = (-3 + √73)/2 or x = (-3 – √73)/2
Leave as a simplified exact answer.
Problem 5: One Solution
Solve:
x² + 10x + 25 = 0
Step 1: Factor
This is a perfect square
(x + 5)(x + 5) = 0
Step 2: Solve
x + 5 = 0 → x = -5
Solution:
x = -5 (only one solution)
Final Thoughts
These best practice problems with solutions show different ways to solve quadratic equations. Each method works for certain problems. Try all of them to see which one feels easiest to you.
