Math Problem Statement

Fully factorize x^2 + 7x + 12 and use your answer to solve x^2 + 7x + 12 = 0.

Solution

Let's solve the problem step by step.


Part (a): Fully factorize x2+7x+12x^2 + 7x + 12

To factorize x2+7x+12x^2 + 7x + 12:

  1. Look for two numbers that multiply to 12 (the constant term) and add up to 7 (the coefficient of xx).

The numbers are 3 and 4 because: 34=12and3+4=73 \cdot 4 = 12 \quad \text{and} \quad 3 + 4 = 7

Thus, the factorization is: x2+7x+12=(x+3)(x+4)x^2 + 7x + 12 = (x + 3)(x + 4)


Part (b): Solve x2+7x+12=0x^2 + 7x + 12 = 0

Using the factorization from part (a): (x+3)(x+4)=0(x + 3)(x + 4) = 0

To solve, set each factor equal to zero: x+3=0x=3x + 3 = 0 \quad \Rightarrow \quad x = -3 x+4=0x=4x + 4 = 0 \quad \Rightarrow \quad x = -4

The solutions are: x=3andx=4x = -3 \quad \text{and} \quad x = -4


Final Answers:

  1. Part (a): (x+3)(x+4)(x + 3)(x + 4)
  2. Part (b): x=3andx=4x = -3 \quad \text{and} \quad x = -4

Would you like further clarification or more examples of similar problems?

Here are 5 related questions to deepen understanding:

  1. How do you factorize quadratic equations with different coefficients for x2x^2?
  2. Can you explain why factorization helps solve quadratic equations?
  3. How would you solve x2+5x+6=0x^2 + 5x + 6 = 0?
  4. What happens if a quadratic equation cannot be factorized neatly?
  5. Can we solve x2+7x+12=0x^2 + 7x + 12 = 0 using the quadratic formula? Would the solutions match?

Tip: Always double-check your factorization by expanding the factors back to verify the original quadratic expression.

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Math Problem Analysis

Mathematical Concepts

Factoring Quadratic Equations
Solving Quadratic Equations

Formulas

Standard form of a quadratic equation: ax^2 + bx + c = 0

Theorems

Zero Product Property

Suitable Grade Level

Grades 7-9