x2-11x+28=0 Factorizacion
Have you ever encountered an equation like this: x² – 11x + 28 = 0? This is a quadratic equation, and solving it can involve different methods. One powerful approach is factoring. This article will guide you through the process of factoring a quadratic equation, using the specific example of x² – 11x + 28 = 0.
Understanding Quadratic Equations
A quadratic equation is an expression of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The variable in this equation is usually represented by “x.” The highest exponent of the variable is 2, hence the term “quadratic.”
There are various ways to solve a quadratic equation, including using the quadratic formula or factoring. Factoring involves rewriting the equation as a product of two linear expressions.
Unveiling the Strategy: Factoring by Grouping
The method we’ll be using to factor our equation is called factoring by grouping. This technique involves rearranging the terms of the equation to group them strategically. Here’s a breakdown of the steps:
Identify the coefficient of the x term (b) and the constant term (c) in the equation. In our case, b = -11 and c = 28.
Find two values (a and d) that satisfy the following conditions:
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- The product of a and d must equal the constant term (c): a * d = c (which is 28 in our example).
- The sum of a and d must equal the coefficient of the x term (b): a + d = b (which is -11 in our example).Rewrite the middle term (bx) of the equation by using the values you found in step 2.Group the terms in the equation such that there’s a common factor in each group. Factor out the common factors from each group. Combine the factored expressions. x2-11x+28=0 factorizacion.
Putting it into Practice: Factoring x² – 11x + 28 = 0
Now, let’s apply the steps mentioned above to factor the equation x² – 11x + 28 = 0:
We already identified b (coefficient of x) as -11 and c (constant term) as 28.
We need to find two values (a and d) that fulfill the conditions:
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- a * d = 28
- a + d = -11
Trying out different factors of 28, we discover that a = 7 and d = 4 satisfy both conditions: 7 * 4 = 28 and 7 + 4 = -11.
We can now rewrite the middle term (-11x) using a and d: -11x = (7x) + (-4x)
Let’s group the terms in the equation: (x² + 7x) + (-4x + 28)
We see a common factor of x in the first group (x² + 7x) and a common factor of -4 in the second group (-4x + 28). Let’s factor them out: x(x + 7) – 4(x – 7)
Finally, we combine the factored expressions: (x – 4)(x + 7)
The Answer Revealed: The Factored Equation
Through the process of factoring by grouping, we’ve been able to rewrite the original equation x² – 11x + 28 = 0 as: (x – 4)(x + 7) = 0
This factored form reveals valuable information. The equation holds true when either of the factors equals zero. Therefore, the solutions to the original equation are x = 4 and x = -7.
Beyond the Example: The Power of Factoring
Factoring quadratic equations is a valuable skill in mathematics. It simplifies equations, aids in solving for unknown variables, and provides insights into the relationships between the variables.
This technique can be applied to various quadratic equations, not just the specific example used in this article. By understanding the steps involved and practicing with different equations, you’ll gain confidence in tackling these mathematical challenges.
For further exploration, you can try factoring quadratic equations with different constant terms and coefficients. There are also other methods for factoring quadratics, such as using the sum-product pattern. With practice and exploration, you’ll develop a strong foundation in solving these equations.
