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Chapter 2: Problem 6

Find the constant term needed to make \(x^{2}+7 x\) a perfect square trinomial.

Short Answer

Expert verified
The necessary constant term is \(49/4\).

Step by step solution

01

Comparing to the general form

To make this a perfect square trinomial, compare \(x^{2}+7 x\) to the general form \(x^{2}+2ax+a^{2}\). The coefficient of 'x' in our equation is 7. Comparing this to the coefficient of 'x' in the general form, we can say that 2a = 7.
02

Finding the value of 'a'

The next step will be to find the value of 'a' by solving the equation 2a = 7. Dividing by 2 gives us \(a=7/2\).
03

Finding the constant term

Now, the term needed to complete the square is \(a^{2}\). So, we calculate \((7/2)^{2}\), which equals \(49/4\). This is the constant term that makes \(x^{2}+7 x\) a perfect square trinomial.

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Key Concepts

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

Completing the Square
The method of 'completing the square' is a cornerstone in solving quadratic equations. It involves transforming a quadratic expression into a perfect square trinomial, which is easier to work with when solving equations. Picture a partially built square block puzzle where you have the base and one side, and your task is to find the exact piece that completes it. Mathematically, this means adding a specific constant term to an expression like \( x^2 + bx \) to form \( (x + p)^2 \).

The key steps include:
  • Finding the coefficient \( b \) of the \'x\' term.
  • Halving \( b \), which gives us \( p \), and then squaring it to get the missing piece.
  • Adding this piece, or constant term, to the original expression to complete the square.
In our exercise, by halving the coefficient 7 and squaring it, we complete the square for the expression \( x^2 + 7x \) by adding \( {(7/2)}^2 \), which is \( 49/4 \).
Algebraic Expressions
Algebraic expressions are the bread and butter of algebra, consisting of numbers, variables (like \( x \) or \( y \)), and operations (such as addition or subtraction). They're like recipes that give instructions on how to mix these ingredients to create new mathematical dishes. For example, \( x^2 + 7x \) tells us to take a variable \( x \), square it, multiply it by 7, and then add these two results.

There are several types of algebraic expressions, including:
  • Monomials (a single term like \( 3x \)).
  • Binomials (two terms like \( x^2 + 7x \)).
  • Polynomials (three or more terms).
Algebraic expressions become really interesting when we manipulate them to reveal something new 鈥� like turning \( x^2 + 7x \) into a perfect square trinomial by completing the square.
Quadratic Equations
Quadratic equations are like puzzles that challenge us to find the value of an unknown variable that makes the equation true. They follow the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. When you see an equation like this, it's asking, 'What number(s) can \( x \) be to make this whole thing equal zero?'

There are several ways to solve these equations, including:
  • Factoring (if the quadratic easily breaks down into binomial factors).
  • Using the quadratic formula (a failsafe method applicable to all quadratics).
  • Graphing (to visually find where the parabola intersects the x-axis).
Completing the square is another technique that's particularly helpful when the equation doesn't factor nicely. It helps us convert the equation into something more manageable, from which we can easily find solutions for \( x \).

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Most popular questions from this chapter

Graph the function \(f(t)=t^{2}-4\) in a decimal window. Using your graph, determine the values of \(t\) for which \(f(t) \geq 0\).

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$f(t)=-t^{2}-1$$

The range of a quadratic function \(g(x)=a x^{2}+b x+c\) is given by \((-\infty, 2] .\) Is \(a\) positive or negative? Justify your answer.

Let \(f(x)=2 x+5\) and \(g(x)=f(x+2)-4 .\) Graph both functions on the same set of coordinate axes. Describe the transformation from \(f(x)\) to \(g(x) .\) What do you observe?

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. $$h(x)=(\sqrt{2}) x^{2}+x+1$$
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