Algebra

x^2 + x - 12 = 0

Step-by-step solution with explanation

Final Answer

and

Step-by-step solution

1

Identify the quadratic to factor

We have a quadratic equation set equal to zero. Factoring is the fastest method here because the numbers are small whole numbers.
2

Find two numbers that multiply and add

For factoring, we need two numbers that multiply to the constant term (-12) and add to the middle coefficient (1). Think through factor pairs of -12 until one pair adds to 1.
3

Identify the correct pair: 4 and -3

✓
The numbers 4 and -3 work perfectly. They multiply to -12 and add to 1, so we can use them to factor the equation.
4

Write the factored form

We split the middle term using our two numbers, 4 and -3. Check by expanding: ✓
5

Apply the Zero Product Property

If two factors multiply to zero, at least one of them must be zero. We set each factor equal to zero and solve separately.
6

Solve each equation for x

Subtract 4 from the first equation to get . Add 3 to the second equation to get . These are both valid solutions.

Understanding this problem

Learning Insight

The Zero Product Property is the key idea: if , then or . This only works when one side equals zero — that's why we always move everything to one side before factoring. Factoring is really just reversing the distributive property (FOIL in reverse).

Quick Tip

To find your factor pair fast, list factor pairs of |c| (here, 12): 1×12, 2×6, 3×4. Then ask which pair has a difference or sum equal to the middle coefficient when you add signs. Here 4 and 3 differ by 1, and we need +1, so the bigger number gets the + sign.

Common Mistake

Students often set the factors equal to zero but then forget to flip the sign — seeing and writing instead of . Remember: whatever is being added to x becomes negative when you solve for x.