Problem 23 - Algebra 1

Question

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}-22 x$$

Step-by-Step Solution

Answer

The term that should be added to the expression \(x^{2} - 22x\) to form a perfect square trinomial is 121.

Step 1: Identify 'a' and '2ab'

For the expression \(x^{2} - 22x + b^{2}\), the coefficient for x squared, which is 1 in this case, is representative of 'a squared'. And, the coefficient for x, which is -22, represents '2ab'. Here, 'a' stands for 1 and '2ab' is -22.

Step 2: Determine 'b'

We've established that 2ab equals -22. Divide -22 by 2 to find 'ab', which is -11. Since 'a' is equal to 1, 'b' is therefore -11.

Step 3: Calculate \(b^{2}\)

Now we need to square 'b' (i.e., find \(b^{2}\)) to find the term to add to the given expression. If 'b' is -11, then \(b^{2} = (-11)^{2} = 121\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is a fundamental skill in Algebra 1. It involves breaking down a quadratic expression, like a trinomial, into simpler binomial factors. Recognizing patterns such as perfect square trinomials and factoring by grouping are common techniques used in this process.
 

• For any quadratic expression in the form of \(ax^2 + bx + c\), the goal is to express it as \((px + q)(rx + s)\).
• The coefficients \(p\), \(q\), \(r\), and \(s\) are determined by specific relationships between \(a\), \(b\), and \(c\).

One special case when factoring quadratic expressions is the perfect square trinomial, where the quadratic can be rewritten as \((x + m)^2\). Identifying these structures allows for efficient factoring, simplifying algebraic operations, and solving equations more easily.

Completing the Square

Completing the square is a method used to solve quadratic equations or to change the form of a quadratic expression. This technique is particularly useful when the goal is to transform a given quadratic into a perfect square trinomial. Here’s how it generally works:

• Given a quadratic expression \(x^2 + bx\), find the value that makes it a perfect square trinomial.
• Halve the linear coefficient (\(b\)), and then square this number. This gives you the term to be added.

In the exercise, by halving -22, we get -11, and squaring -11 results in 121. So, adding 121 to \(x^2 - 22x\) forms the perfect square trinomial \((x - 11)^2\).
This method is also instrumental in deriving the quadratic formula and is widely used in calculus and higher-level mathematics courses.

Algebra 1 Concepts

In Algebra 1, quadratic expressions and equations are pivotal concepts. They set the foundation for solving problems and understanding relationships between variables. Key components taught in this segment include linear equations, factoring, functions, and graphing parabola shapes.

• The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), provides solutions to any quadratic equation.
• Understanding the symmetry and vertex of parabolas is crucial when graphing or solving quadratics.

Through mastering basics like factoring and completing the square, students can more clearly see the connections between algebraic concepts, preparing them for subsequent math courses. These initial topics allow students to tackle diverse math problems with confidence and competence.