Algebra & Functions · Grades 9–11
How to Solve Quadratic Equations: Factoring and the Formula
Step 1: get the equation into standard form
Move every term to one side so the other side is 0, then read off a, b and c with their signs. x² = 3x + 4 becomes x² − 3x − 4 = 0, so a = 1, b = −3 and c = −4.
Step 2: try factoring first
Look for two numbers that multiply to c and add to b. For x² − 3x − 4, those numbers are −4 and +1, giving (x − 4)(x + 1). Setting each bracket to zero gives the two roots.
Step 3: use the formula when factoring is not obvious
The formula works for every quadratic. Compute the discriminant D = b² − 4ac first; it tells you what kind of answer to expect.
- D > 0 → two different real roots
- D = 0 → one repeated real root
- D < 0 → no real roots; two complex roots
x² − 3x − 4 = 0
- Factored form: (x − 4)(x + 1)
- Discriminant D = 25
- Vertex: (1.5, −6.25)
Step-by-step working 6 steps
- 1Identify a, b and c
x² − 3x − 4 = 0 a = 1, b = −3, c = −4
- 2Calculate the discriminant
D = b² − 4ac = (−3)² − 4(1)(−4) = 25
D is positive: two different real roots.
- 3Apply the quadratic formula
x = (−b ± √D) / 2a = (3 ± √25) / 2 = (3 ± 5) / 2
- 4Solve for both roots
x₁ = 4 x₂ = −1
- 5Check by substitution
f(4) = 0 f(−1) = 0
Both should be 0 (allowing for tiny rounding).
- 6Vertex of the parabola
x = −b / 2a = 1.5, y = −6.25 → vertex (1.5, −6.25)
The parabola opens upward, so the vertex is the minimum point.
When a is not 1
Factoring is harder when a ≠ 1, but the formula does not care. 2x² + 3x − 2 = 0 gives x = 0.5 or x = −2.
2x² + 3x − 2 = 0
- Factored form: 2(x − 0.5)(x + 2)
- Discriminant D = 25
- Vertex: (−0.75, −3.125)
Step-by-step working 6 steps
- 1Identify a, b and c
2x² + 3x − 2 = 0 a = 2, b = 3, c = −2
- 2Calculate the discriminant
D = b² − 4ac = (3)² − 4(2)(−2) = 25
D is positive: two different real roots.
- 3Apply the quadratic formula
x = (−b ± √D) / 2a = (−3 ± √25) / 4 = (−3 ± 5) / 4
- 4Solve for both roots
x₁ = 0.5 x₂ = −2
- 5Check by substitution
f(0.5) = 0 f(−2) = 0
Both should be 0 (allowing for tiny rounding).
- 6Vertex of the parabola
x = −b / 2a = −0.75, y = −3.125 → vertex (−0.75, −3.125)
The parabola opens upward, so the vertex is the minimum point.
When there are no real roots
If D is negative, the square root in the formula is of a negative number. The roots are complex: x² + 2x + 5 = 0 gives x = −1 ± 2i. Graphically, the parabola never touches the x-axis.
Negative discriminant
- No real x-intercepts: the parabola never crosses the x-axis.
- Discriminant D = −16
- Vertex: (−1, 4)
Step-by-step working 4 steps
- 1Identify a, b and c
x² + 2x + 5 = 0 a = 1, b = 2, c = 5
- 2Calculate the discriminant
D = b² − 4ac = (2)² − 4(1)(5) = −16
D is negative: no real roots — two complex roots.
- 3Complex roots
x = (−2 ± √16·i) / 2 x = −1 ± 2i
The square root of a negative number introduces i, where i² = −1.
- 4Vertex of the parabola
x = −b / 2a = −1, y = 4 → vertex (−1, 4)
The parabola opens upward, so the vertex is the minimum point.
Common mistakes
Losing the sign of b
In the formula the first term is −b. If b is already negative, −b is positive.
Dividing only the square root by 2a
Both −b and the root are divided by 2a.
Skipping the check
Substitute each root into the original equation. Both should give 0.
Frequently asked questions
Can a quadratic have just one solution?
Yes, when the discriminant is exactly zero, the parabola just touches the x-axis and the single root is counted twice.
Is the quadratic formula the same as completing the square?
The formula is what you get when you complete the square on the general equation. They are two routes to the same answer.
