Problem 6 - Algebra 1

Question

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}-14 x$$

Step-by-Step Solution

Answer

The term that should be added to form a perfect square trinomial is 49.

Step 1: Identify the first two terms of the trinomial

Identify the first two terms from the given expression which are \(x^{2}\) and \(-14 x\).

Step 2: Find the square of half of the coefficient of the second term

The coefficient of the second term is -14. Its half is -7. The square of -7 is 49.

Step 3: Identify the third term of the trinomial

The third term that should be added to obtain a perfect square trinomial becomes \(49\).

Key Concepts

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is extremely useful in solving quadratic equations, and involves creating an expression that readily factors into a binomial squared.

• The first step in completing the square is to take the quadratic expression and focus on the quadratic and linear terms.


• Take the coefficient of the linear term, divide it by two, and then square it. This result becomes the term needed to complete the square. In the case of the expression \(x^{2}-14x\), the coefficient of \(x\) is \(-14\). Dividing \(-14\) by two gives \(-7\), and squaring \(-7\) results in \(49\).


• This computed square becomes the third term necessary to form the perfect square trinomial \((x-7)^{2}\), which equals \(x^{2}-14x+49\).

Completing the square not only aids in transforming expressions but also helps in identifying the vertex form of quadratic equations. This technique is instrumental in graphing parabolas and solving problems related to motion and optimization.

Quadratic Expressions

Quadratic expressions are mathematical expressions of the form \(ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These expressions can represent parabolic graphs and are foundational in algebra.

• The main components of a quadratic expression are the quadratic term \(ax^{2}\), the linear term \(bx\), and the constant term \(c\).


• When graphed, quadratic expressions form parabolas, which can open upwards or downwards depending on the sign of the coefficient \(a\).


• The vertex of the parabola is a critical point that can be found through completing the square or using the vertex formula \(x = -\frac{b}{2a}\).

Quadratic expressions are pivotal in algebraic operations, including factoring, simplifying, and solving quadratic equations. Knowing how to handle them is essential for tackling more complex algebraic problems and real-world applications such as projectile motion.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and coefficients assembled through operations such as addition, subtraction, multiplication, and division. They are the building blocks of algebra, providing a way to represent mathematical ideas and relationships.

• Variables represent unknown values, while constants are fixed numbers.


• Coefficients are numbers that multiply the variables, indicating how much of the variable is present.


• Operations combine these elements to form expressions that can vary from simple monomials (like \(3x\)) to complex polynomials (like \(5x^{3} - 2x^{2} + 3x - 4\)).

Understanding algebraic expressions is crucial as they enable the formulation and solution of problems using symbols. This understanding creates a bridge to more advanced mathematical fields and lays the groundwork for exploring equations, functions, and inequalities.