Factorizing quadratic expressions is a crucial technique in algebra with countless applications. One such expression is x^{2} + 12x + 35, understanding how to factor it completely is a valuable skill. This comprehensive guide will provide a systematic and detailed explanation of the steps involved in factoring x^{2} + 12x + 35.

To begin, it’s important to understand the concept of factoring. Factoring involves breaking down an expression into a product of simpler expressions. In the case of x^{2} + 12x + 35, the goal is to express it as a product of two linear factors in the form of (ax + b)(cx + d). By finding the appropriate values of a, b, c, and d, we can factor x^{2} + 12x + 35 completely.

The process of factoring x^{2} + 12x + 35 requires identifying two numbers that multiply to 35 and add to 12. These numbers are 7 and 5. Using this information, we can rewrite x^{2} + 12x + 35 as x^{2} + 7x + 5x + 35.

## Grouping

### Grouping Similar Terms

To proceed further, we group similar terms together: (x^{2} + 7x) and (5x + 35).

### Factoring Out the Greatest Common Factor

We can factor out the greatest common factor from each group: x from the first and 5 from the second. This gives us (x(x + 7)) and (5(x + 7)).

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## Combining the Factors

### Combining the Parentheses

Since we have the same factor (x + 7) in both terms, we can combine them: (x + 7)(x + 5).

### The Final Factored Form

Therefore, the complete factorization of x^{2} + 12x + 35 is **(x + 7)(x + 5)**.

## Applications

### Solving Quadratic Equations

Factoring completely is essential for solving quadratic equations. For example, to solve the equation x^{2} + 12x + 35 = 0, we first factor it as (x + 7)(x + 5). Then, we set each factor equal to zero and solve for x: x + 7 = 0 and x + 5 = 0. This gives us the solutions x = -7 and x = -5.

### Completing the Square

Factoring completely is also used in completing the square, a technique for transforming a quadratic expression into a perfect square. This is useful for finding the vertex of a parabola and solving quadratic equations without using the quadratic formula.

### Graphing Quadratics

The factored form of a quadratic expression provides valuable information about its graph. The x-intercepts correspond to the solutions of the equation and the vertex can be determined from the factored form.

## Additional Notes

### Checking the Solution

To verify if the factorization is correct, multiply the factors back together: (x + 7)(x + 5) = x^{2} + 12x + 35. Since this matches the original expression, the factorization is correct.

### Special Cases

In some cases, a quadratic expression may not be factorable over the real numbers. This occurs when the discriminant (b^{2} – 4ac) is negative. For example, x^{2} + 12x + 36 cannot be factored over the real numbers because its discriminant is 0.

### Conclusion

Factoring completely x^{2} + 12x + 35 is a fundamental skill in algebra with numerous applications. By following the steps outlined in this guide, you can master the technique and confidently solve a wide range of quadratic problems.

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