Problem 45 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 9 x^{2}-36 x-45 $$

Step-by-Step Solution

Answer

The factored form of \(9x^2-36x-45\) is \(3(x+1)(x-5)\).

Step 1: Identify a Common Factor

Observe the terms in the quadratic expression: \(9x^2\), \(-36x\), and \(-45\). Notice that all terms are divisible by 3. The common factor is 3. Factor 3 out of the entire expression: \[ 9x^2 - 36x - 45 = 3(3x^2 - 12x - 15). \]

Step 2: Simplify Factorable Quadratic

Now, focus on the expression inside the parentheses: \(3x^2 - 12x - 15\). We will factor this as a product of two binomials. It can be expressed as a form of \(ax^2 + bx + c\), where \(a = 3\), \(b = -12\), and \(c = -15\).

Step 3: Find Two Numbers that Multiply and Add to Specific Values

We need two numbers that multiply to \(a \times c = 3 \times (-15) = -45\) and add to \(b = -12\). These numbers are \(-15\) and \(3\) since \(-15 \times 3 = -45\) and \(-15 + 3 = -12\).

Step 4: Substitute and Factor by Grouping

Rewrite \(-12x\) as \(-15x + 3x\) in the expression. Therefore:\[ 3x^2 - 12x - 15 = 3x^2 - 15x + 3x - 15. \] Group the terms:\[ (3x^2 - 15x) + (3x - 15). \] Factor out the greatest common factor from each group:\[ 3x(x - 5) + 3(x - 5). \] Notice that \(x-5\) is a common factor, factor it out:\[ (3x + 3)(x - 5). \]

Step 5: Simplify the Expression

Further simplify by factoring out the common factor of 3 from \(3x + 3\): \[ 3(x + 1)(x - 5). \]

Step 6: Combine with Initial Factor

Finally, combine this with the initial factor of 3 we factored out in Step 1:\[ 3(x + 1)(x - 5) = 3(x + 1)(x - 5). \] This is the completely factored form of the original expression \(9x^2 - 36x - 45\).

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial that can be written in the form of \( ax^2 + bx + c \). This means it is an equation where the highest exponent of the variable \( x \) is 2. Quadratic expressions appear frequently in algebra and are important for various mathematical concepts. For example, the equation \( 9x^2 - 36x - 45 \) is a quadratic expression where \( a = 9 \), \( b = -36 \), and \( c = -45 \).
The challenge often involves rewriting the expression into a simpler form, such as the product of two binomials. This simplification can make solving equations easier and illuminate relationships between variables.

Common Factor

A common factor in mathematics refers to a number that divides into each term of the expression without leaving a remainder. Identifying the common factor is a crucial step in simplifying expressions.
In the case of the expression \( 9x^2 - 36x - 45 \), the common factor is 3. Each term in the expression—\( 9x^2 \), \(-36x \), and \(-45 \)—is divisible by 3. By factoring out this common factor, the expression is simplified to \[ 3(3x^2 - 12x - 15) \].
This step is important as it simplifies the expression, making it easier to work with when continuing with further methods like factoring by grouping.

Binomial

A binomial is a polynomial expression containing exactly two terms. Examples include \( x + 1 \) and \( x - 5 \). When factoring quadratic expressions, one aim is to express the quadratic as a product of two binomials.
Finding these binomials involves identifying numbers that satisfy certain conditions based on the expression's coefficients. These conditions are typically derived from the constants \( a \), \( b \), and \( c \) of the quadratic form \( ax^2 + bx + c \). This process is crucial in turning quadratic expressions into simplified, factorable forms like \((x + 1)(x - 5)\).

Factoring by Grouping

Factoring by grouping is a method used to factor expressions that are not initially easy to simplify into binomials. This approach is useful when dealing with quadratic expressions after factoring out a common factor.
Consider the expression \( 3x^2 - 12x - 15 \). To factor by grouping, rewrite the middle term \(-12x\) into two groups: \(-15x\) and \( 3x \). Then rewrite the expression as \( (3x^2 - 15x) + (3x - 15) \).
Next, extract the greatest common factor from each group: \(3x(x - 5) + 3(x - 5)\). Notice both groups share a common binomial factor of \( (x - 5) \). Factoring this out further reduces the expression to \((3x + 3)(x - 5)\). Simplify further to complete the factorization. This method highlights the utility of grouping terms to find commonalities and simplify complex expressions.