2x Squared Minus x Squared: Simplifying Polynomial Expressions

In this section, we will simplify the expression \(2x^2 - x^2\). This is a basic algebraic operation that involves combining like terms.

Steps to Simplify

1. Identify the like terms: Both \(2x^2\) and \(x^2\) are like terms because they contain the variable \(x\) raised to the same power.
2. Combine the coefficients of the like terms:
   • Subtract the coefficient of \(x^2\) from the coefficient of \(2x^2\).
   • This gives us: \(2x^2 - x^2 = (2 - 1)x^2 = x^2\).
3. Write the simplified expression: The simplified form of \(2x^2 - x^2\) is \(x^2\).

Mathematical Representation

Using MathJax, the simplification process can be written as:

\[
2x^2 - x^2 = (2 - 1)x^2 = x^2
\]

Example Usage

This simplification is useful in various mathematical problems and equations. For instance, when solving quadratic equations or simplifying polynomial expressions, recognizing and combining like terms is a fundamental skill.

Additional Resources

For more information on simplifying algebraic expressions and solving quadratic equations, you can explore the following resources:

Simplifying the expression \(2x^2 - x^2\) involves combining like terms to achieve a simplified form. Here's a step-by-step guide to the process:

• Identify the terms in the expression: \(2x^2\) and \(-x^2\).
• Since both terms are like terms (they have the same variable raised to the same power), we can combine them by subtracting the coefficients.
• Subtract the coefficient of \(-x^2\) from the coefficient of \(2x^2\):
  • \(2x^2 - x^2 = (2 - 1)x^2 = 1x^2 = x^2\)

Therefore, the simplified form of \(2x^2 - x^2\) is \(x^2\).

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. These equations can be solved using various methods, with factoring being one of the primary approaches.

Factoring Quadratic Equations

Factoring involves expressing the quadratic equation as a product of two binomial expressions. Here are the detailed steps to factor a quadratic equation:

1. Write the equation in standard form: Ensure the quadratic equation is in the form \( ax^2 + bx + c = 0 \).
2. Identify coefficients: Recognize the coefficients \( a \), \( b \), and \( c \).
3. Find two numbers that multiply to \( ac \) and add to \( b \): This is often the trickiest part. For example, for \( 6x^2 + 5x - 6 \), \( ac = -36 \) and \( b = 5 \). The numbers -4 and 9 satisfy these conditions because \( -4 \times 9 = -36 \) and \( -4 + 9 = 5 \).
4. Rewrite the middle term: Split the middle term using the two numbers found. For instance, rewrite \( 6x^2 + 5x - 6 \) as \( 6x^2 - 4x + 9x - 6 \).
5. Group terms and factor each group: Factor out common factors in pairs of terms. Here, \( 6x^2 - 4x + 9x - 6 \) becomes \( 2x(3x - 2) + 3(3x - 2) \).
6. Factor out the common binomial: Combine the factored groups to get \( (2x + 3)(3x - 2) \).

Solving Factored Equations

Once the quadratic equation is factored, solving it involves setting each binomial factor to zero and solving for \( x \):

• For \( (2x + 3) = 0 \), \( x = -\frac{3}{2} \).
• For \( (3x - 2) = 0 \), \( x = \frac{2}{3} \).

Thus, the solutions to \( 6x^2 + 5x - 6 = 0 \) are \( x = -\frac{3}{2} \) and \( x = \frac{2}{3} \).

The graphical interpretation of the expression \(2x^2 - x^2\) involves simplifying the expression and understanding its graphical representation. Simplifying \(2x^2 - x^2\) gives us \(x^2\), which is a standard quadratic function. Here, we will explore the steps to graph this quadratic function and interpret its features.

Steps to Graph \(x^2\)

1. Identify the basic form: The function \(x^2\) is a simple quadratic function with the general form \(ax^2 + bx + c\), where \(a = 1\), \(b = 0\), and \(c = 0\).
2. Vertex: For \(x^2\), the vertex is at the origin \((0,0)\) because there are no linear or constant terms to shift the graph horizontally or vertically.
3. Axis of symmetry: The axis of symmetry for \(x^2\) is the y-axis, which is the line \(x = 0\).
4. Plotting points: Calculate and plot several points around the vertex. For example:
   • When \(x = -2\), \(y = (-2)^2 = 4\)
   • When \(x = -1\), \(y = (-1)^2 = 1\)
   • When \(x = 0\), \(y = 0^2 = 0\)
   • When \(x = 1\), \(y = 1^2 = 1\)
   • When \(x = 2\), \(y = 2^2 = 4\)
5. Draw the parabola: Connect the plotted points with a smooth curve to form a U-shaped parabola opening upwards.

Features of the Graph

• Vertex: The lowest point on the graph at \((0, 0)\).
• Axis of symmetry: The line \(x = 0\) (the y-axis) divides the parabola into two mirror-image halves.
• Intercepts: The graph intersects the y-axis at the origin \((0, 0)\) and the x-axis also at the origin, indicating that the only root of \(x^2 = 0\) is \(x = 0\).
• Direction: The parabola opens upwards, indicating that the function \(x^2\) is always non-negative for all real values of \(x\).
• End behavior: As \(x\) approaches positive or negative infinity, \(y\) increases without bound.

Advanced algebraic techniques are essential for solving complex mathematical problems. These techniques involve a deep understanding of polynomial manipulation, factoring, and solving equations. Let's explore some of these advanced methods step by step.

• Polynomials

Understanding polynomials is crucial for advanced algebra. Polynomials are algebraic expressions that consist of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents of variables. The standard form of a polynomial is written as:
$$ a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 $$

• Factoring Polynomials

Factoring is breaking down a complex expression into simpler components or factors that, when multiplied together, give the original expression. For instance, to factor \(2x^2 - x^2\), we simplify it to:
$$ x^2 (2 - 1) = x^2 $$
Factoring can involve recognizing patterns like the difference of squares, perfect square trinomials, or using techniques such as grouping.

• Using the Distributive Property

The distributive property is a fundamental property of multiplication over addition or subtraction. It is expressed as:
$$ a(b + c) = ab + ac $$
This property is essential for expanding and simplifying algebraic expressions.

• Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form \(x - c\). It is generally used for polynomials of higher degrees and provides an efficient way to find roots or simplify expressions.

• Solving Quadratic Equations

Quadratic equations are polynomials of degree 2 and can be solved using various methods like factoring, completing the square, or applying the quadratic formula:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Each method provides a different approach depending on the specific form of the quadratic equation.

• Logarithms and Exponents

Logarithms and exponents are inverses of each other and are fundamental in solving exponential equations. Understanding the properties and rules of exponents and logarithms, such as the power rule, product rule, and quotient rule, is essential for manipulating expressions involving these functions.

• Rationalizing Expressions

Rationalizing involves rewriting a fraction so that the denominator contains only rational numbers. This technique is often used when the denominator includes radicals. For example:
$$ \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$
Rationalizing helps simplify expressions and makes them easier to work with.

• Understanding the expression \(2x^2 - x^2\) involves basic polynomial operations, particularly simplification by combining like terms.
• Real-world scenarios often require simplifying algebraic expressions to model situations or solve problems efficiently.
• For instance, in physics, when calculating motion equations involving acceleration and velocity terms, simplifying expressions like \(2x^2 - x^2\) helps in deriving accurate formulas.
• In economics, understanding polynomial simplification aids in optimizing cost functions or analyzing revenue models.
• Moreover, in engineering disciplines such as structural analysis, polynomial operations are crucial for formulating design equations or solving systems of equations.