Problem 98 - Algebra A Combined Function

Question

Fill in the blank so that \(9 x^{2}+_________x+25\) is a perfect square trinomial.

Step-by-Step Solution

Answer

Fill in the blank with 30.

Step 1: Recognize the form of a perfect square trinomial

A perfect square trinomial is generally expressed as \(a^2 + 2ab + b^2\). This implies that for the expression to be a perfect square, its form should be similar to \( (3x)^2 + 2ab + b^2\) where 3x is equivalent to a.

Step 2: Identify the given components

We know that \(9x^2\) is \(a^2\) and can be rewritten as \((3x)^2\), meaning \(a = 3x\). Similarly, \(25\) is \(b^2\) and can be rewritten as \(5^2\), meaning \(b = 5\).

Step 3: Determine the missing term

The expression for a perfect square trinomial includes a middle term \(2ab\). Substitute the values \(a = 3x\) and \(b = 5\) into this formula: \(2ab = 2(3x)(5)\). Simplify to find the middle term: \(2ab = 30x\).

Step 4: Complete the expression

Insert the value found for the missing term in the blank: \(9x^2 + 30x + 25\). This new expression is now a perfect square trinomial.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They provide a way to describe mathematical relationships in a concise form. Although algebraic expressions don't have an equals sign, unlike equations, they are crucial in forming and solving equations. For example, in the expression \(9x^2 + 30x + 25\), each part plays a role.

• \(9x^2\) is the quadratic term where 9 is the coefficient, and \(x\) is the variable raised to the second power.
• \(30x\) is a linear term, consisting of the coefficient 30 and the variable \(x\)
• \(25\) is a constant term, representing a fixed number.

These components can change based on the values of the variables, making algebraic expressions highly dynamic and flexible in solving various mathematical problems.

Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\). They represent parabolas when plotted on a graph. Quadratic equations are vital in various real-world applications such as physics, engineering, and economics.
Key Characteristics of Quadratic Equations

• The highest exponent is 2, which indicates the 'quadratic' nature.
• They can have up to two solutions or roots.
• Solutions can be found using various methods like factoring, completing the square, and using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

Perfect square trinomials, like our example \(9x^2 + 30x + 25 = (3x+5)^2\), represent a special form of quadratic equations where the squared expression simplifies the solving process.

Trinomial Factoring

Trinomial factoring is the process of breaking down a quadratic trinomial expression into the product of two binomials. It is an essential skill in algebra, aiding in simplifying equations and solving for variables. To factor a perfect square trinomial, such as the one found: \(9x^2 + 30x + 25\), identify it expressed as \((a + b)^2\).
Factors of a Perfect Square Trinomial

• Determine the square roots of the first and last terms.
• Check if the middle term is twice the product of those square roots.
• If so, the trinomial can be represented in an expanded binomial square form like \((3x + 5)^2\).

Factoring allows the identification of critical points, making it easier to solve or graph quadratic equations effectively. Understanding trinomial factoring is a fundamental step in mastering algebra and solving more complex mathematical problems.