Learn / Solve a Quadratic Equation Step by Step
Solve a Quadratic Equation Step by Step
A quadratic equation can be written as ax² + bx + c = 0. First move every term to one side so the other side is 0, then pick a method: factoring is fastest when it works, and the quadratic formula always works.
For: Algebra 1 and Algebra 2 students (grades 8 to 11).
The rules
- Write the equation in standard form: ax² + bx + c = 0.
- Factoring: find two numbers that multiply to a·c and add to b, split the middle term using those two numbers, factor by grouping, then set each factor equal to 0.
- Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a).
- The discriminant b² − 4ac tells you how many real solutions: positive = 2, zero = 1, negative = none.
Worked examples
Example 1: Solve x² + 5x + 6 = 0 by factoring
- Find two numbers that multiply to 6 and add to 5: 2 and 3.
- Factor: (x + 2)(x + 3) = 0.
- Set each factor to 0: x + 2 = 0 or x + 3 = 0.
Answer: x = −2 or x = −3
Example 2: Solve 2x² − 3x − 2 = 0 with the quadratic formula
- a = 2, b = −3, c = −2.
- Discriminant: (−3)² − 4(2)(−2) = 9 + 16 = 25.
- x = (3 ± √25) / 4 = (3 ± 5) / 4.
- (3 + 5)/4 = 2 and (3 − 5)/4 = −1/2.
Answer: x = 2 or x = −1/2
Example 3: Solve x² − 6x + 4 = 0 by completing the square
- Move the constant: x² − 6x = −4.
- Half of −6 is −3, and (−3)² = 9. Add 9 to both sides: x² − 6x + 9 = 5.
- Write as a square: (x − 3)² = 5.
- Take square roots: x − 3 = ±√5.
Answer: x = 3 ± √5
Example 4: Solve x² = 3x + 10
- Standard form: x² − 3x − 10 = 0.
- Two numbers that multiply to −10 and add to −3: −5 and 2.
- (x − 5)(x + 2) = 0.
Answer: x = 5 or x = −2
Practice problems
- x² − 7x + 12 = 0
- x² − 16 = 0
- 3x² + 2x − 1 = 0
- x² + 4x + 1 = 0
- x² + 2x + 5 = 0
Show answers
- x = 3 or x = 4
- x = 4 or x = −4
- x = 1/3 or x = −1
- x = −2 ± √3
- No real solutions (discriminant −16)
Frequently asked questions
Which method should I use to solve a quadratic?
Try factoring first if the numbers are small. If it does not factor nicely, use the quadratic formula, which works for every quadratic.
Can a quadratic have no solution?
It can have no real solution. That happens when the discriminant b² − 4ac is negative.
Stuck on your own problem? Type it in and Layla, Algegram's AI tutor, shows every step. Your first 5 questions are free, no sign-up.
Ask my math question →