Algebra

x^2 + 5x + 6 = 0

Step-by-step solution with explanation

Final Answer

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Step-by-step solution

1

Write out the equation clearly

We start with the quadratic equation as given. Our goal is to find all values of x that make this equation true.
2

Find two numbers that multiply and add

To factor a quadratic, we need two numbers that multiply to the constant term (6) and add to the middle coefficient (5). Think through pairs: 1×6=6 and 1+6=7 (no), 2×3=6 and 2+3=5 (yes!).
3

Factor the quadratic expression

We rewrite the quadratic as a product of two binomials using the numbers 2 and 3. You can check by expanding: ✓
4

Apply the Zero Product Property

If two things multiply to equal zero, then at least one of them must be zero. So we set each factor equal to zero separately.
5

Solve each equation for x

Subtract 2 from both sides of the first equation to get x = -2. Subtract 3 from both sides of the second to get x = -3. These are our two solutions.

Understanding this problem

Learning Insight

Factoring works because of the Zero Product Property — a fundamental rule that says if A × B = 0, then A = 0 or B = 0. This only works when one side equals zero, which is why we always move everything to one side before factoring.

Quick Tip

To quickly find the factor pair, list factor pairs of the constant term (6): (1,6) and (2,3). Check which pair adds up to the middle coefficient (5). The pair 2 and 3 works instantly!

Common Mistake

Students often set the factors equal to zero but then forget to flip the sign — writing x = 2 and x = 3 instead of x = -2 and x = -3. Remember: if x + 2 = 0, you subtract 2, giving x = -2, not +2.