Math Problem Statement

Factorise x^2 + 7x + 12 and solve x^2 + 7x + 12 = 0.

Solution

Let's go through the problem step-by-step:

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

We need to factorize the quadratic expression x2+7x+12x^2 + 7x + 12.

  1. Look for two numbers that multiply to 12 and add up to 7. These numbers are 3 and 4, since:

    • 3×4=123 \times 4 = 12
    • 3+4=73 + 4 = 7
  2. Rewrite the expression as a product of two binomials: x2+7x+12=(x+3)(x+4)x^2 + 7x + 12 = (x + 3)(x + 4)

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

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

To solve the equation x2+7x+12=0x^2 + 7x + 12 = 0, we can use the factors we found in part (a).

  1. Set each factor equal to zero:

    • x+3=0x + 3 = 0 or x+4=0x + 4 = 0
  2. Solve for xx in each case:

    • x=3x = -3
    • x=4x = -4

So, the solutions to the equation x2+7x+12=0x^2 + 7x + 12 = 0 are: x=3andx=4x = -3 \quad \text{and} \quad x = -4

Would you like more details, or do you have any other questions?


Related Questions to Explore:

  1. How can we factorize other quadratic expressions with different coefficients?
  2. What if the quadratic expression cannot be factorized easily—how else can we solve it?
  3. How does the quadratic formula apply to this problem?
  4. Why does factoring work as a method for solving quadratic equations?
  5. Can these methods apply to higher-degree polynomials?

Tip:

When factoring quadratics, look for factor pairs that match both the product and sum conditions for quick factorization.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring Quadratics
Solving Quadratic Equations

Formulas

Factoring by finding factors that add to the middle term and multiply to the constant term

Theorems

Zero Product Property

Suitable Grade Level

Grades 7-9