Factorizing quadratic expressions is a crucial technique in algebra with countless applications. One such expression is x2 + 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 x2 + 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 x2 + 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 x2 + 12x + 35 completely.

The process of factoring x2 + 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 x2 + 12x + 35 as x2 + 7x + 5x + 35.

Grouping

Grouping Similar Terms

To proceed further, we group similar terms together: (x2 + 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)).

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 x2 + 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 x2 + 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) = x2 + 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 (b2 – 4ac) is negative. For example, x2 + 12x + 36 cannot be factored over the real numbers because its discriminant is 0.

Conclusion

Factoring completely x2 + 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.

Tags:

Share:

Related Posts :

Leave a Comment