Understanding the Expression: x² + 5x + 6 + x²
Breaking Down the Expression
The given algebraic expression is:
```plaintext
x² + 5x + 6 + x²
```
At first glance, it appears to be a combination of quadratic terms, linear terms, and a constant. Let's analyze each component:
- Quadratic terms: x² and x²
- Linear term: 5x
- Constant term: 6
Since the expression contains two x² terms, they can be combined to simplify the expression.
Simplification of the Expression
Combining like terms is crucial in algebra. For this expression:
```plaintext
x² + x² + 5x + 6
```
which simplifies to:
```plaintext
2x² + 5x + 6
```
This is a quadratic expression in its simplified form. The coefficient of x² is 2, which indicates the parabola opens upwards and is wider than the standard parabola y = x².
Factorization of the Quadratic Expression
Factoring 2x² + 5x + 6
Factoring quadratic expressions is a key skill in algebra. The goal is to express the quadratic as a product of binomials:
```plaintext
( ax + b )( cx + d )
```
such that:
```plaintext
a c = coefficient of x² (here, 2)
b d = constant term (here, 6)
a d + b c = coefficient of x (here, 5)
```
Let's find factors step-by-step.
Step 1: Find pairs of factors of 6 (constant term):
- 1 and 6
- 2 and 3
Step 2: Find factors of 2 (coefficient of x²):
- 1 and 2
Now, test combinations to satisfy the middle term 5x:
- (2x + 3)(x + 2):
Expand: 2x x + 2x 2 + 3 x + 3 2 = 2x² + 4x + 3x + 6 = 2x² + 7x + 6 (not matching 5x)
- (2x + 1)(x + 6):
Expand: 2x x + 2x 6 + 1 x + 1 6 = 2x² + 12x + x + 6 = 2x² + 13x + 6 (not matching)
- (x + 2)(2x + 3):
Expand: x 2x + x 3 + 2 2x + 2 3 = 2x² + 3x + 4x + 6 = 2x² + 7x + 6 (not matching)
- (x + 3)(2x + 2):
Expand: x 2x + x 2 + 3 2x + 3 2 = 2x² + 2x + 6x + 6 = 2x² + 8x + 6 (not matching)
- (2x + 1)(x + 6):
Already considered; results in 13x, not 5x.
- (2x + 3)(x + 2):
Already considered; results in 7x, not 5x.
Alternative approach: Use the quadratic formula or completing the square if factoring proves difficult.
Step 3: Use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a=2\), \(b=5\), \(c=6\).
Calculate discriminant:
\[
\Delta = 5^2 - 4 2 6 = 25 - 48 = -23
\]
Since the discriminant is negative, the quadratic has complex roots, thus it cannot be factored over real numbers.
Conclusion: The quadratic is prime over the real numbers, but can be factored over complex numbers.
Applications of the Expression in Algebra and Beyond
Graphing the Quadratic Function
The simplified form \( y = 2x^2 + 5x + 6 \) represents a parabola opening upward. Key features include:
- Vertex: The highest or lowest point of the parabola.
- Axis of symmetry: The vertical line passing through the vertex.
- Y-intercept: The point where the parabola crosses the y-axis.
To find the vertex:
\[
x_{v} = -\frac{b}{2a} = -\frac{5}{2 2} = -\frac{5}{4}
\]
Calculate y-coordinate:
\[
y_{v} = 2\left(-\frac{5}{4}\right)^2 + 5\left(-\frac{5}{4}\right) + 6
\]
\[
= 2 \times \frac{25}{16} - \frac{25}{4} + 6
\]
\[
= \frac{50}{16} - \frac{25}{4} + 6
\]
\[
= \frac{25}{8} - \frac{25}{4} + 6
\]
Express all with denominator 8:
\[
= \frac{25}{8} - \frac{50}{8} + \frac{48}{8} = \frac{25 - 50 + 48}{8} = \frac{23}{8}
\]
So, the vertex is at:
\[
\left( -\frac{5}{4}, \frac{23}{8} \right)
\]
Y-intercept:
Set \(x=0\):
\[
y = 2(0)^2 + 5(0) + 6 = 6
\]
X-intercepts:
Set \(y=0\):
\[
2x^2 + 5x + 6 = 0
\]
Discriminant \(\Delta = -23\), negative, so no real roots; the parabola does not cross the x-axis.
Real-World Applications
Quadratic expressions like \(2x^2 + 5x + 6\) are used in various fields:
- Physics: Modeling projectile motion, where path equations are quadratic.
- Economics: Calculating profit functions, which can be quadratic.
- Engineering: Analyzing stress and strain in materials.
- Biology: Modeling population dynamics with quadratic growth or decay.
Related Algebraic Concepts
Completing the Square
Since the quadratic cannot be factored over reals, completing the square offers an alternative method:
\[
2x^2 + 5x + 6
\]
Divide entire expression by 2:
\[
x^2 + \frac{5}{2}x + 3
\]
Complete the square:
\[
x^2 + \frac{5}{2}x + \left( \frac{5}{4} \right)^2 - \left( \frac{5}{4} \right)^2 + 3
\]
\[
= \left( x + \frac{5}{4} \right)^2 - \frac{25}{16} + 3
\]
Express 3 as \(\frac{48}{16}\):
\[
= \left( x + \frac{5}{4} \right)^2 + \frac{23}{16}
\]
This form reveals the vertex directly and is useful for solving quadratic equations or analyzing the graph.
Quadratic Equation Solutions
When solving quadratic equations, methods include:
- Factoring (if possible)
- Completing the square
- Quadratic formula
Given the discriminant is negative, solutions are complex:
\[
x = \frac{-5 \pm \sqrt{-23}}{4}
\]
Expressed in complex form:
\[
x = -\frac{5}{4} \pm \frac{\sqrt{23}}{4} i
\]
Summary and Conclusion
The algebraic expression x² + 5x + 6 + x² simplifies to 2x² + 5x + 6, a quadratic polynomial with a positive leading coefficient. Its analysis involves understanding its structure, roots, graph, and applications across different fields. Although it does not factor over the real numbers, completing the square and the quadratic formula provide insight into its properties and solutions.
Understanding such expressions is essential for students
Frequently Asked Questions
What is the simplified form of the expression x^2 + 5x + 6?
The expression x^2 + 5x + 6 can be factored into (x + 2)(x + 3).
How do you factor the quadratic expression x^2 + 5x + 6?
To factor x^2 + 5x + 6, find two numbers that multiply to 6 and add to 5, which are 2 and 3, so the factors are (x + 2)(x + 3).
What are the roots of the quadratic equation x^2 + 5x + 6 = 0?
The roots are x = -2 and x = -3, obtained by setting each factor equal to zero.
Can the expression x^2 + 5x + 6 be written as a product of binomials?
Yes, it can be written as (x + 2)(x + 3).
What is the significance of the factors (x + 2) and (x + 3) in the quadratic expression?
They represent the factors corresponding to the roots of the quadratic, indicating where the expression equals zero.
How is the quadratic expression x^2 + 5x + 6 related to its factors?
The quadratic can be expressed as the product of its factors: (x + 2)(x + 3), which helps in solving equations and analyzing the graph.
What method can be used to factor the quadratic x^2 + 5x + 6?
You can use factoring by inspection, trial and error, or the quadratic formula, but factoring by inspection is straightforward here.
What is the value of the quadratic expression x^2 + 5x + 6 when x = 1?
Substituting x = 1, the value is 1^2 + 5(1) + 6 = 1 + 5 + 6 = 12.
