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Chapter 2: Problem 5

Find the constant term needed to make \(x^{2}-6 x\) a perfect square trinomial.

Short Answer

Expert verified
The constant term needed to make \(x^{2}-6 x\) a perfect square trinomial is 9.

Step by step solution

01

Identifying the form and a value

First, compare the given expression \(x^{2}-6 x\) to a perfect square trinomial pattern \(a^{2} - 2ab + b^{2}\). Here, 'a' is obviously 'x', because \(x^{2}\) is the first term of the given polynomial and also first term of the perfect square trinomial pattern.
02

Determining 2ab and b value

In a perfect square trinomial, the middle term, which is -6x in this case, is equal to 2ab. We know a = x and we are to find b. The expression -6x represents -2ab, so we can solve for b by dividing -6x by -2a, which is -2x: \(b= -6x/-2x = 3\). So, the b value needed to form a perfect square is b = 3.
03

Determining \(b^{2}\)

Once b is found in Step 2, we need to square it (calculate \(b^{2}\)) because in a perfect square trinomial the last term is \(b^{2}\). Here, \(b^{2} = (3)^2 = 9\). So, the constant term needed to form a perfect square trinomial is 9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expressions
Polynomial expressions are algebraic expressions that contain variables, coefficients, and exponents. These expressions can have one or multiple terms, and each term is composed of:
  • A coefficient, which is a constant number.
  • Variables, such as x or y, raised to a non-negative integer exponent.
For example, in the expression \(3x^2 - 5x + 2\), there are three terms: \(3x^2\), \(-5x\), and \(2\). The degree of a polynomial is determined by the highest exponent in the expression, which in this example is 2. Therefore, it鈥檚 a quadratic polynomial. Understanding the basic structure of polynomials helps in classifying them and in performing operations like addition, subtraction, and multiplication.
Quadratic Expressions
Quadratic expressions are a subset of polynomial expressions where the highest power of the variable is 2. They usually take the standard form \(ax^2 + bx + c\) where:
  • \(a\), \(b\), and \(c\) are coefficients.
  • \(a eq 0\) to ensure there is a quadratic term.
A key characteristic of quadratic expressions is their ability to represent parabolic graphs when plotted. They can take on different forms such as:
  • General form: \(ax^2 + bx + c\)
  • Vertex form: \(a(x-h)^2 + k\)
  • Factored form: \(a(x-r)(x-s)\)
Each representation provides unique insights into the properties of the quadratic function, like its vertex or roots. Mastering quadratic expressions is crucial as they form the foundation for topics like solving quadratic equations and graphing parabolas.
Completing the Square
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly helpful for solving quadratic equations and for deriving the quadratic formula. The process involves:
  • Rewriting the quadratic expression in the form of \(ax^2 + bx\).
  • Taking the coefficient of the linear term \(b\), dividing it by 2, and then squaring it to find the constant term \(c\).
  • Adding and subtracting this constant term \(c\) within the expression to maintain equality.
For instance, with the expression \(x^2 - 6x\), take half of -6, which is -3, and square it to get 9. Hence, adding 9 completes the perfect square trinomial, transforming it into \((x-3)^2 = x^2 - 6x + 9\). This method simplifies many algebraic processes and provides a pathway to solve equations by observing the expression鈥檚 simpler square form.

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Most popular questions from this chapter

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=3 x-1$$

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=(-2 x+5)^{2}$$

Graph each quadratic function by finding a suitable viewing window with the help of the TABLE feature of a graphing utility. Also find the vertex of the associated parabola using the graphing utility. $$h(x)=x^{2}+5 x-20$$

A chartered bus company has the following price structure. A single bus ticket costs \(\$ 30 .\) For each additional ticket sold to a group of travelers, the price per ticket is reduced by \(\$ 0.50 .\) The reduced price applies to all the tickets sold to the group. (a) Calculate the total cost for one, two, and five tickets. (b) Using your calculations in part (a) as a guide, find a quadratic function that gives the total cost of the tickets. (c) How many tickets must be sold to maximize the revenue for the bus company?

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\frac{1}{x^{2}+1} ; g(x)=\frac{2 x+1}{3 x-1}$$
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