X 2 2x 35 Factored

Factoring the Expression x² + 2x - 35: A thorough look

This article provides a complete walkthrough on factoring the quadratic expression x² + 2x - 35. We will explore various methods for solving this problem, delving into the underlying mathematical principles and offering practical examples to solidify your understanding. Factoring quadratic expressions is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding more advanced mathematical concepts. Mastering this skill will significantly enhance your problem-solving abilities in mathematics.

Understanding Quadratic Expressions

Before diving into the factoring process, let's establish a solid understanding of what a quadratic expression is. It generally takes the form ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (usually x) is 2. In our case, the quadratic expression is x² + 2x - 35, where a = 1, b = 2, and c = -35.

Method 1: Factoring by Finding Factors

This method involves finding two numbers that add up to the coefficient of the x term (b) and multiply to the constant term (c). Let's apply this to our expression x² + 2x - 35:

Identify b and c: In our expression, b = 2 and c = -35.
Find two numbers that add to b and multiply to c: We need two numbers that add up to 2 and multiply to -35. After some trial and error (or a systematic approach, explained below), we find that 7 and -5 satisfy these conditions: 7 + (-5) = 2 and 7 * (-5) = -35.
Rewrite the expression: We can rewrite the original expression using these two numbers: x² + 7x - 5x - 35.
Factor by grouping: Now, we group the terms in pairs and factor out the common factors: x(x + 7) - 5(x + 7)
Factor out the common binomial: Notice that (x + 7) is a common factor in both terms. We can factor it out: (x + 7)(x - 5).

So, the factored form of x² + 2x - 35 is (x + 7)(x - 5).

Method 2: The Quadratic Formula

The quadratic formula is a more general method that can be used to find the roots (or zeros) of any quadratic equation of the form ax² + bx + c = 0. These roots can then be used to factor the quadratic expression. The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

Let's apply this to our expression:

Identify a, b, and c: a = 1, b = 2, c = -35
Substitute into the quadratic formula:

x = [-2 ± √(2² - 4 * 1 * -35)] / (2 * 1) x = [-2 ± √(4 + 140)] / 2 x = [-2 ± √144] / 2 x = [-2 ± 12] / 2

Solve for the two roots:

x₁ = (-2 + 12) / 2 = 5 x₂ = (-2 - 12) / 2 = -7

Construct the factored form: The roots of the quadratic equation correspond to the factors. If x₁ = 5, then one factor is (x - 5). If x₂ = -7, then the other factor is (x + 7). That's why, the factored form is (x - 5)(x + 7).

Method 3: Trial and Error (Systematic Approach)

The trial-and-error method can be made more efficient with a systematic approach. Since the coefficient of x² is 1, we look for two numbers that multiply to -35 and add up to 2. We list the factor pairs of -35:

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1 and -35
-1 and 35
5 and -7
-5 and 7

Only the pair -5 and 7 adds up to 2. Because of this, the factors are (x - 5) and (x + 7).

Mathematical Explanation: Why Factoring Works

The distributive property of multiplication is the fundamental principle behind factoring. When we expand (x + 7)(x - 5), we use the FOIL method (First, Outer, Inner, Last):

First: x * x = x²
Outer: x * -5 = -5x
Inner: 7 * x = 7x
Last: 7 * -5 = -35

Combining these terms, we get x² - 5x + 7x - 35 = x² + 2x - 35, which is our original expression. Factoring reverses this process, breaking down the expression into its multiplicative components.

Solving Quadratic Equations using Factoring

Once we've factored the expression (x + 7)(x - 5), we can use it to solve the quadratic equation x² + 2x - 35 = 0. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore:

x + 7 = 0 => x = -7
x - 5 = 0 => x = 5

The solutions to the equation are x = -7 and x = 5. These are the roots or zeros of the quadratic equation.

Frequently Asked Questions (FAQ)

Q: What if the coefficient of x² is not 1?

A: If the coefficient of x² is not 1, the factoring process becomes slightly more complex. You might need to use techniques like the AC method or grouping, where you find factors that satisfy both the sum of the coefficients and the product of the coefficients multiplied by the leading coefficient.

Q: What if the quadratic expression cannot be factored easily?

A: If the quadratic expression cannot be easily factored, you can always use the quadratic formula to find the roots and then construct the factored form using the roots.

Q: Are there other methods for factoring quadratic expressions?

A: Yes, there are other advanced techniques, such as completing the square and using the difference of squares formula, but the methods outlined above are sufficient for most basic quadratic expressions.

Q: Why is factoring important?

A: Factoring is a fundamental skill in algebra that allows you to simplify expressions, solve equations, and understand the behaviour of quadratic functions. It's a building block for more advanced mathematical concepts.

Conclusion

Factoring the quadratic expression x² + 2x - 35 is a straightforward process once you understand the underlying principles. Day to day, this factored form provides valuable insights into the solutions of the corresponding quadratic equation and forms the basis for further algebraic manipulations. Mastering this skill is crucial for progressing in algebra and related mathematical fields. Whether you use the method of finding factors, the quadratic formula, or a systematic trial-and-error approach, the result is the same: (x + 7)(x - 5). Worth adding: remember to practice regularly to build your fluency and confidence in tackling similar problems. The more you practice, the easier it will become to recognize patterns and efficiently solve these types of problems.