Problem 66 - Elementary Algebra

Question

\(c^{2}+7 c=0\)

Step-by-Step Solution

Answer

The solutions are \[ c = 0 \] and \[ c = -7 \].

Step 1 - Identify the Equation Type

Recognize that the given equation is a quadratic equation of the form ewline\[ c^2 + 7c = 0 \].

Step 2 - Factor the Equation

Factor out the common factor. The equation can be rewritten as: ewline\[ c(c + 7) = 0 \].

Step 3 - Set Each Factor Equal to Zero

To find the solutions, set each factor equal to zero: ewline\[ c = 0 \]ewline and ewline\[ c + 7 = 0 \].

Step 4 - Solve Each Equation

Solve each equation separately: ewlineFor ewline\[ c = 0 \], the solution is ewline\[ c = 0 \].ewlineFor ewline\[ c + 7 = 0 \], subtract 7 from both sides to get ewline\[ c = -7 \].

Step 5 - State the Solutions

The solutions to the quadratic equation are: ewline\[ c = 0 \] and ewline\[ c = -7 \].

Key Concepts

factoring

Factoring is an essential algebraic process in solving equations. It's all about breaking down an expression into simpler expressions (called factors) that, when multiplied together, give the original expression.
In the context of quadratic equations like \[ c^2 + 7c = 0 \], factoring helps us transform the equation into a product of simpler binomials. For example:
\[ c^2 + 7c = c(c + 7) = 0 \]
Here, we factor out the common factor \( c \). This process simplifies our quadratic equation, making it easier to solve.
The most common factors to look for include:

• Common factors shared by all terms
• Differences of squares
• Perfect square trinomials

Mastering factoring is crucial because it is a stepping stone to more complex algebraic manipulations.

solving quadratic equations

When solving quadratic equations, our goal is to find the values of the variable that make the equation true. Quadratic equations take the general form \[ ax^2 + bx + c = 0 \]. Solving them typically involves:
- Factoring the quadratic equation
- Using the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Completing the square

In our specific problem \[ c^2 + 7c = 0 \, \], we started by recognizing it as a quadratic equation. We then factored it to get:
\[ c(c + 7) = 0 \].
Once factored, the solutions come from setting each factor equal to zero and solving for the variable. This could involve simple operations, such as adding or subtracting constants. Mastering these techniques allows you to handle all forms of quadratic equations efficiently.

zero product property

The Zero Product Property is a fundamental principle in algebra. It states that if the product of two numbers is zero, then at least one of the multiplicands must be zero.
Mathematically, if \[ ab = 0 \], then either \[ a = 0 \] or \[ b = 0 \] (or both).
In the context of solving quadratic equations, after factoring them, we apply this property to find solutions. For the equation \[ c(c + 7) = 0 \, \] the Zero Product Property tells us that either \[ c = 0 \] or \[ c + 7 = 0 \. \]
So we solve:

• \[ c = 0 \]
• \[ c + 7 = 0 \] (which simplifies to \[ c = -7 \])

This property is powerful because it turns a more complex equation into simpler linear equations, making it easier for us to solve.

basic algebra

Understanding basic algebra is crucial for solving quadratic equations and many other math problems. Basic algebra includes operations like addition, subtraction, multiplication, and division, along with more complex operations such as exponentiation and root extraction.
Some key concepts include:

• Combining like terms
• Distributive property
• Solving linear equations
• Factoring expressions

For the given quadratic equation \[ c^2 + 7c = 0 \], basic algebra provides the skills needed to factor and solve it. We use distributive property for factoring:
\[ c^2 + 7c = c(c + 7) = 0 \]
Then, we solve the simpler linear equations derived from factoring. Having a solid grasp on basic algebra makes understanding and completing these steps straightforward. Thus, investing time in learning and practicing basic algebra lays a strong foundation for tackling more advanced mathematical problems.