Which shows the factored form of x2 12x 45 – The factorization of x2 + 12x + 45 opens a gateway into the realm of algebraic manipulation, revealing a technique that empowers us to break down complex expressions into simpler forms. This guide will delve into the steps, significance, and applications of factoring, providing a comprehensive understanding of this fundamental mathematical operation.
By mastering the art of factoring, we unlock a powerful tool that enables us to solve equations, simplify expressions, and gain insights into the behavior of quadratic functions. Throughout this discourse, we will explore the intricacies of factoring, unraveling its mysteries and illuminating its practical applications in the real world.
Factoring the Expression x2 + 12x + 45
Factoring is a mathematical technique that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on factoring the expression x 2+ 12x + 45.
Steps for Factoring
To factor the expression x 2+ 12x + 45, we can use the following steps:
- Find two numbers that add up to 12 (the coefficient of the x-term) and multiply to 45 (the constant term).
- These numbers are 9 and 5.
- Rewrite the middle term, 12x, as the sum of these two numbers, 9x + 5x.
- Factor by grouping:
- (x2+ 9x) + (5x + 45)
- Factor out the greatest common factor from each group:
- x(x + 9)
- 5(x + 9)
- Combine the two factors:
- (x + 9)(x + 5)
Significance of the Factored Form
The factored form of x 2+ 12x + 45, which is (x + 9)(x + 5), provides valuable insights into the behavior of the expression.
- Zeroes:The zeroes of the expression are the values of x that make it equal to zero. By setting each factor to zero, we find that the zeroes are x = -9 and x = -5.
- Intercepts:The intercepts are the points where the graph of the expression crosses the x- and y-axes. The x-intercepts are (-9, 0) and (-5, 0), and the y-intercept is (0, 45).
- Shape of the Graph:The factored form shows that the expression is a parabola that opens upward. The vertex of the parabola is (-7, 10), which is the minimum value of the expression.
Applications of Factoring
Factoring has numerous applications in real-world problems, including:
- Solving Equations:Factoring can be used to solve quadratic equations by setting the expression equal to zero and then factoring it to find the zeroes.
- Simplifying Expressions:Factoring can simplify complex expressions by breaking them down into simpler factors.
- Finding the Least Common Multiple (LCM) and Greatest Common Factor (GCF):Factoring can help find the LCM and GCF of two or more expressions.
- Geometric Applications:Factoring is used in geometry to find the area, perimeter, and volume of various shapes.
Extensions, Which shows the factored form of x2 12x 45
In addition to the method described above, there are other methods for factoring quadratic expressions, such as:
- Completing the Square:This method involves completing the square of the x-term and then factoring the resulting expression.
- Quadratic Formula:The quadratic formula can be used to find the zeroes of a quadratic expression without factoring it.
Factoring is closely related to solving quadratic equations. By factoring the expression, we can easily find the zeroes of the equation and determine its behavior.
FAQ Explained: Which Shows The Factored Form Of X2 12x 45
What is the factored form of x2 + 12x + 45?
The factored form of x2 + 12x + 45 is (x + 9)(x + 5).
How do you factor x2 + 12x + 45?
To factor x2 + 12x + 45, you can use the following steps: 1) Find two numbers that add up to 12 and multiply to 45. These numbers are 9 and 5. 2) Rewrite the middle term as the sum of these two numbers: 12x = 9x + 5x.
3) Factor by grouping: (x + 9)(x + 5).
What are the applications of factoring?
Factoring has numerous applications, including solving equations, simplifying expressions, and finding the roots of quadratic functions. It is also used in physics, engineering, and economics to model and solve real-world problems.
