Problem 47 - Elementary Algebra

Question

Solve. $$1 x-4+x x-2=2 x 2-6 x+8$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{-3 + \sqrt{65}}{2} \) and \( x = \frac{-3 - \sqrt{65}}{2} \).

Step 1: Distribute Terms

Let's start by distributing terms on both sides of the equation. On the left side, distribute the variables: \( 1 \cdot x - 4 + x \cdot x - 2 = x - 4 + x^2 - 2 \) Stay aware of any like terms which could be combined later.

Step 2: Simplify Both Sides

Now, let's simplify each side of the equation. Simplify the left side: - Combine like terms: \[ x + x^2 - 4 - 2 = x^2 + x - 6 \] Next, simplify the right side: - Distribute and combine terms: \[ 2 \cdot 2x - 6x + 8 = 4x - 6x + 8 = -2x + 8 \] The equation now is: \[ x^2 + x - 6 = -2x + 8 \]

Step 3: Move All Terms to One Side

To solve the equation, we first move all terms to one side to set it to zero. - Move all terms from the right side to the left: \[ x^2 + x - 6 + 2x - 8 = 0 \] - Combine like terms: \[ x^2 + (x + 2x) - 6 - 8 = 0 \] \[ x^2 + 3x - 14 = 0 \]

Step 4: Solve the Quadratic Equation

Now, solve the quadratic equation \( x^2 + 3x - 14 = 0 \). We can factor or use the quadratic formula. Let's use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) Here, \( a = 1 \), \( b = 3 \), and \( c = -14 \). - Calculate \( b^2 - 4ac \): \[ 3^2 - 4(1)(-14) = 9 + 56 = 65 \] - Apply the quadratic formula: \[ x = \frac{-3 \pm \sqrt{65}}{2} \] Therefore, the solutions are \( x = \frac{-3 + \sqrt{65}}{2} \) and \( x = \frac{-3 - \sqrt{65}}{2} \).

Key Concepts

Factoring Quadratics

Factoring quadratics can be a powerful tool for solving quadratic equations. A quadratic equation is typically expressed in the form \( ax^2 + bx + c = 0 \). If we can express the quadratic equation as a product of two binomials, we can use factoring to find solutions for \( x \).
To factor a quadratic:

• Look for two numbers that multiply to \( ac \) (the product of the coefficient of \( x^2 \) and the constant term).
• These numbers should also add up to \( b \) (the coefficient of \( x \)).
• Rearrange the quadratic expression into two binomial factors and solve each for \( x \).

If factoring appears too complex or impossible due to unfriendly numbers, the quadratic formula provides an alternative solution method without needing to factor.

Quadratic Formula

The quadratic formula is an invaluable tool for solving any quadratic equation, especially when factoring isn’t straightforward. It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). This formula provides a method to find the precise solutions to any quadratic equation.Here's how to use it:

• First, calculate the discriminant, \( b^2 - 4ac \), to check the number of solutions (real or complex).
• Use the formula to solve for \( x \) by substituting \( a \), \( b \), and \( c \).
• The \( \pm \) symbol indicates two potential solutions: add the square root term in one instance and subtract it in the other.

The value of the discriminant informs the nature of solutions:

• A positive discriminant implies two distinct real solutions.
• Zero results in one real (repeated) solution.
• A negative discriminant indicates two complex solutions.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions, like those often seen on either side of an equation. Like terms are terms within an expression that have identical variable parts. This means they include the same variable, raised to the same power.To combine:

• Identify terms that are "like." For instance, \( x^2 \) terms match with other \( x^2 \) terms, and constant numbers can combine with other constants.
• Add or subtract the coefficients of these like terms.
• Simplify the expression to make solving easier.

Example: In the expression \( x^2 + 3x^2 - 4x + 5x - 7 \), combine the \( x^2 \) terms to get \( 4x^2 \), and the \( x \) terms to get \( x \). The simplified form is \( 4x^2 + x - 7 \). This process helps in maintaining an organized path toward the solution.

Distributive Property

The distributive property is a fundamental algebraic property used to simplify expressions. It involves multiplying a single term by each term within a parenthesis, ensuring that multiplication affects all components.Formally, it’s expressed as:\[ a(b + c) = ab + ac \]This simplifies expression by "distributing" multiplication over addition or subtraction within the brackets.
Steps to apply the distributive property:

• Identify the term outside the parenthesis that needs to be distributed to each term inside.
• Multiply this term across all terms inside the parenthesis.
• Write the expression without parenthesis, after completing the multiplication.

In a quadratic equation, you might distribute to form expressions like \( x(x + 2) \) into \( x^2 + 2x \). This process simplifies the expression and is an essential first step before further manipulation and solving the equation.