Problem 38 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ c^{2}+10 c-48=12 c $$

Step-by-Step Solution

Answer

The solutions to the equation are \(c = 8\) and \(c = -6\).

Step 1: Organize the equation

First, we need to organize the equation. Because it's easier to identify the coefficients if they're in the general form, organize the equation to look like \(c^{2}+bx-constant=0\). We can do this by subtracting \(12c\) from both sides. So, \(c^{2}+10c-12c-48=0\)

Step 2: Simplify the Equation

Next, simplify the equation by combining like terms. The \(c\) terms get added together, so the equation becomes \(c^{2}-2c-48=0\)

Step 3: Factoring the equation.

After rearranging the equation, we will factor the quadratic expression on the right. The expression \(c^{2}-2c-48\) factors into \((c-8)(c+6)=0\)

Step 4: Solve for c

We can now see that our equation \((c-8)(c+6)=0\) will be satisfied when \((c-8) = 0\) , giving a solution of \(c = 8\), and when \((c+6) = 0\) , giving a solution of \(c = -6\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that has a degree of 2. In simple terms, this means the highest exponent of the variable (often denoted as \( x \) or \( c \)) is 2. Quadratic equations take the general form: \[ ax^2 + bx + c = 0 \] Here, \( a \), \( b \), and \( c \) are known as coefficients, and the equation represents a parabola graphically. Some key points to note about quadratic equations are:

• The equation can have up to two real solutions, known as roots or zeroes.
• The solutions can be found using various methods such as factoring, completing the square, or using the quadratic formula.
• The discriminant \( b^2 - 4ac \) helps determine the nature of the roots (real vs. imaginary).

Understanding how to recognize and manipulate these equations is crucial for solving them effectively. In our example, the quadratic equation \( c^2 - 2c - 48 = 0 \) is obtained by rearranging and simplifying the original equation.

Solving Equations

To solve quadratic equations, several strategies can be employed. Factoring is often the simplest approach if the equation can be easily factorized. The process typically involves:

• Rearranging the equation into the standard quadratic form \( c^2 + bx + c = 0 \).
• Identifying two numbers that multiply to the constant term and add up to the middle term's coefficient.
• Using these numbers to express the quadratic expression as a product of two binomials.

In our example, the quadratic expression \( c^2 - 2c - 48 \) is factored into \( (c - 8)(c + 6) = 0 \). This factorization is useful because it transforms the equation into a simpler form, where each factor can be set to zero to find the solutions. This leads to the possible solutions: * \( c = 8 \) * \( c = -6 \) Solving these two simpler equations gives the values of \( c \) that satisfy the original quadratic equation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( x \) or \( c \)), and operations (such as addition and multiplication). They form the foundation of algebraic equations and are used to represent mathematical relationships. In the case of quadratic equations, understanding how to manipulate these expressions is key to solving the equations.
Consider the expression \( c^2 - 2c - 48 \). This is a quadratic expression because it contains the term \( c^2 \), which defines its degree. Key aspects include:

• **Like Terms**: These are parts of the expression that contain the same variable raised to the same power. For example, \( 10c - 12c \) simplifies to \(-2c \).
• **Factoring**: The process of breaking down a complex expression into simpler, multiply-able factors. For instance, the given expression \( c^2 - 2c - 48 \) can be expressed as \((c - 8)(c + 6)\).

Being adept at simplifying and factoring algebraic expressions can greatly ease the process of finding solutions to quadratic equations. Recognizing patterns and relationships within these expressions enhances one's problem-solving prowess.