Question

What are the values of \(a\) in this equation? $$ 3 a^{2}-27 a-108=0 $$ (A) \(-9,-3\) (B) \(6,-4\) (C) 9,6 (D) \(12,-3\)

Short answer

The values of \(a\) are 12 and -3, matching option (D).

Step-by-step solution

Identify the Quadratic Equation

The given equation is in the form of a standard quadratic equation: \(3a^2 - 27a - 108 = 0\), where \(a\) is the variable. We'll solve for \(a\) to find its values.

Simplify the Coefficients

Notice that each term in the equation can be divided by 3. Let's simplify it: \(a^2 - 9a - 36 = 0\). This makes it easier to factor.

Factor the Quadratic Equation

We need to find two numbers that multiply to \(-36\) (the constant term) and add to \(-9\) (the coefficient of \(a\)). The numbers \(-12\) and \(3\) satisfy these conditions. Thus, we can factor the equation as: \((a - 12)(a + 3) = 0\).

Solve for \(a\)

Set each factor of the equation equal to zero: \(a - 12 = 0\) and \(a + 3 = 0\). Solving these gives \(a = 12\) and \(a = -3\).

Match with Given Options

The solutions of the equation are \(a = 12\) and \(a = -3\). This matches option (D).

Key concepts

Factoring Quadratic Equations

Factoring quadratic equations involves expressing the equation in a product of simpler terms called factors. Consider the simplified equation from the original example, which is: \[a^2 - 9a - 36 = 0\]We need to identify two numbers that multiply to the constant term, which is \(-36\), and add up to the coefficient of \(a\), which is \(-9\). Finding these numbers requires a bit of trial and error or systematic analysis. In this case, the numbers \(-12\) and \(+3\) satisfy these conditions. This means we can rewrite the quadratic equation as a product of two binomials: \[(a - 12)(a + 3) = 0\] By expressing it this way, we have factored the quadratic equation. This step is crucial because it allows us to solve for the variable more easily by setting each factor to zero.

Solving Quadratic Equations

Solving quadratic equations typically involves finding the values of the variable that satisfy the equation. Once we have the equation factored, like \[ (a - 12)(a + 3) = 0 \], we can solve for the unknown variable \(a\) by setting each factor equal to zero.

• First, set \( (a - 12) = 0\). Solving this gives: \ a = 12 \.
• Next, set \( (a + 3) = 0\). Solving this gives: \ a = -3 \.

These values, \( a = 12 \) and \ a = -3 \, are the solutions to the equation. We've identified the points where the quadratic expression equals zero, often referred to as the roots or zeros of the equation.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations. In the context of quadratic equations, we use algebraic methods to simplify, factor, and solve the equations.The original equation given was \( 3a^2 - 27a - 108 = 0 \), which is a typical example seen in algebra classes. To handle this using algebra, the first step was simplifying by dividing every term by the largest common factor, which was 3 in this case. This step simplifies the coefficients, making the equation \( a^2 - 9a - 36 = 0 \), thereby making the factoring process less complex.Algebra provides tools to transition equations into forms that reveal important properties, like the roots. It involves a clear understanding of operations and the relationships between quantities, often transforming complex problems into simpler ones that can be easily managed.