Problem 51 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ 3 a^{2}=12 $$

Step-by-Step Solution

Answer

Question: Solve the equation $$3a^2 = 12$$ and find the possible values for a. Answer: The possible values for 'a' are 2 and -2.

Step 1: Make 'a' term the subject of the equation

To achieve this, we'll divide both sides of the equation by 3: $$ \frac{3a^2}{3} = \frac{12}{3} $$ This simplifies to: $$ a^2 = 4 $$

Step 2: Take the square root of both sides

We now take the square root of both sides to isolate 'a': $$ \sqrt{a^2} = \pm \sqrt{4} $$

Step 3: Solve for 'a'

Now, we calculate the possible value(s) for 'a': $$ a = \pm 2 $$ The possible values for 'a' are 2 and -2, and thus we have found the solution to the given equation.

Key Concepts

Square Root Method

The square root method is a straightforward way to solve quadratic equations, particularly when the equation is in the form of \(a^2 = b\). This method is useful because it involves isolating the squared variable and then taking its square root. Here’s a step-by-step guide:

• Isolate the squared term: You need to make sure that the variable squared is by itself on one side of the equation. For the equation given, \(a^2 = 4\), 'a' is already isolated.


• Apply the square root: To solve for 'a', take the square root of both sides of the equation. Remember, the square root of a number can be both positive and negative. Therefore, taking the square root of \(4\), we get \(a = \pm 2\).

By following these steps, you can efficiently solve equations of this type and discover the possible values of the variable involved.

Quadratic Equation

Quadratic equations are algebraic expressions of the form \(ax^2 + bx + c = 0\). However, in simpler cases like \(3a^2 = 12\), the equation is reduced, potentially missing linear (b) or constant (c) terms.
Even though these expressions may appear simple, they use the fundamental principle of quadratics.In quadratics where you set one side equal to zero, you apply various methods such as:

• Factoring: Breaking the equation into multiplied binomials.
• Completing the square: Rewriting it as a perfect square trinomial.
• Quadratic formula: Using the formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to find solutions, especially for complex equations.

Each approach is a tool in your algebraic toolkit, enabling you to tackle different quadratic scenarios efficiently.

Algebraic Equations

Algebraic equations involve finding values for variables that make the equation true. Variables are symbols, typically letters, used to represent unknown numbers.
These equations can range from simple linear forms like \(x + 2 = 5\) to more complex quadratic expressions.When solving algebraic equations, the main goal is to isolate the variable. Key steps include:

• Simplification: Combine like terms and remove any excess constants.
• Isolation of variables: Through addition, subtraction, multiplication, or division, manipulate the equation to keep the variable on one side.
• Verification: Substitute the found value back into the original equation to ensure it holds true.

Mastering these algebraic steps ensure you're well-prepared to handle any equation type, enhancing your problem-solving skills greatly.