Problem 65 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ 3 a^{2}+18 a+27=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation $$3a^2 + 18a + 27 = 0$$ if possible and find its roots. Answer: The quadratic equation can be factored into $$(a + 3)(a + 3) = 0$$, and its unique, double root is \(a = -3\).

Step 1: Check if the quadratic equation can be factored

Begin by looking for common factors in the coefficients of the equation and check if the equation can be factored. The given equation is: $$3a^2 + 18a + 27 = 0$$. The common factor in all the coefficients is 3. Divide the equation by this common factor: $$a^2 + 6a + 9 = 0$$Now, try to find two numbers whose product is 9 and sum is 6. In this case, we can see that 3 * 3 = 9 and 3 + 3 = 6. So, we can factor the equation as: $$(a + 3)(a + 3) = 0$$

Step 2: Find the roots of the factored equation

Now that the equation has been factored, we just need to find the roots. The factored equation is: $$(a + 3)(a + 3) = 0$$To find the roots, set each factor equal to zero and solve for 'a': 1. \(a + 3 = 0\) Subtract 3 from both sides: \(a = -3\) Since both of the factors are \((a + 3)\), there is only one root: \(a = -3\). This is a repeated or double root, meaning that the quadratic equation touches the x-axis at this unique point. The solution to the given quadratic equation is \(a = -3\).

Key Concepts

Factoring Quadratics

Understanding how to factor quadratics is akin to solving a puzzle. The key is to express the quadratic equation, typically written in the form of \( ax^2 + bx + c = 0 \), as a product of two binomials. The process of factoring quadratics involves rewriting the original equation in the form \( (x + m)(x + n) = 0 \), where \( m \) and \( n \) are numbers when multiplied give \( c \) (the constant term), and when added, equal \( b \) (the coefficient of the linear term).

In our original exercise \( 3a^2 + 18a + 27 = 0 \), we seek to find such a pair \( m \) and \( n \) that satisfy these conditions. First, recognize the common factor - in this case, 3 - and factor it out. The resulting simpler quadratic \( a^2 + 6a + 9 = 0 \) cleverly hides the products of \( (a + 3)(a + 3) \). Recognizing this duplication, we conclude that the quadratic factors into a perfect square.

When students encounter difficulty factoring, helpful tips include listing out possible factor pairs of the constant term and checking which pair sums up to the linear coefficient. It's like breaking a code where the product and sum lead directly to the solution.

Finding Roots of Quadratic Equation

Finding the roots of a quadratic equation, also known as solving the equation, is the next logical step after factoring. A root, or zero, of the equation, is a value that, when substituted into the original equation, yields zero. It represents the point(s) at which the graph of the quadratic crosses or touches the x-axis.

As illustrated in the solution, once we have the factored form \( (a + 3)(a + 3) = 0 \), our task simplifies. We now apply the Zero Product Property, which states that if a product of two factors equals zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for \( a \: \, a + 3 = 0 \). In this case, solving \( a + 3 = 0 \) yields the repeated root \(-3\).

The concept of repeated or double roots is vital; it indicates that the graph is just tangent to the x-axis at that point and does not cross it. This phenomenon affects the shape and the vertex of the parabola formed by the quadratic equation.

Algebraic Expressions

Algebraic expressions are mathematical phrases that involve numbers, variables, and arithmetic operations. A quadratic equation itself is an algebraic expression set to zero. When dealing with such expressions, one manipulates algebraic terms to simplify or solve equations.

Take our original problem. The step of dividing by the common factor of 3 transforms \( 3a^2 + 18a + 27 = 0 \) into a more manageable expression \( a^2 + 6a + 9 = 0 \: \), which is still an algebraic expression. Enhancing our algebraic fluency allows us to more readily identify the structure of the expression, like recognizing a perfect square trinomial, which in turn facilitates the factoring process.

It’s crucial to become comfortable not just with executing operations, but also with understanding the structure and nuances of different algebraic expressions to skillfully navigate through various problem-solving scenarios.