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Chapter 7: Problem 8

In Exercises \(5-14\) , determine whether the expression on the left of the equal sign is a difference of squares or a perfect square trinomial. If is, indicate which and then factor the expression and solve the equation for \(x\) . If the expression is in neither form, say so. $$ x^{2}-14 x+49=0 $$

Short Answer

Expert verified
The expression is a perfect square trinomial, factored as \( (x - 7)^2 \), and the solution is \( x = 7 \).

Step by step solution

01

- Identify the Form of the Expression

Observe the expression \( x^{2} - 14x + 49 = 0 \). Determine if it is a difference of squares or a perfect square trinomial. Notice that it has three terms and can be compared to the form \( a^2 - 2ab + b^2 \), which defines a perfect square trinomial.
02

- Compare with the Standard Form

Rewrite the expression in the standard form: \( x^2 - 2*7*x + 7^2 = 0 \). Here, \( a = x \) and \( b = 7 \). This confirms that the expression is a perfect square trinomial because it fits the form \( (a - b)^2 \).
03

- Factor the Expression

Factor the perfect square trinomial using the form \( (a - b)^2 \). So, \( x^2 - 14x + 49 = (x - 7)^2 \). Now the equation is \( (x - 7)^2 = 0 \).
04

- Solve for \( x \)

To solve for \( x \), set the factor equal to zero: \( (x - 7)^2 = 0 \). This implies \( x - 7 = 0 \). Therefore, solving for \( x \) gives \( x = 7 \).

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Key Concepts

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

Difference of Squares
The 'difference of squares' is an important concept in algebra, where a quadratic expression can be written as the difference between two perfect squares. A general expression in this form is \[a^2 - b^2\] and can be factored as \[(a + b)(a - b)\]. To identify an expression as a difference of squares:
  • Ensure it has only two terms.
  • Confirm that each term is a perfect square.
  • Check that the terms are subtracted from one another.
For instance, consider \[x^2 - 49\]. Here,\[a = x\] and \[b = 7\] because:
  • \[x^2\] is a perfect square.
  • \[49 = 7^2\] is a perfect square.
Thus, this can be written and factored as \[(x + 7)(x - 7)\].
This method is quick and helps simplify solving quadratic equations.
Perfect Square Trinomials
A 'perfect square trinomial' refers to a quadratic expression that can be written in the squared form \[(a - b)^2\] or \[(a + b)^2\]. These take the forms: \[a^2 - 2ab + b^2\] and \[a^2 + 2ab + b^2\]. To identify a perfect square trinomial, look for three terms fitting these patterns.
For example, consider \[x^2 - 14x + 49\]. We notice:
  • The first term is a perfect square, \[x^2\].
  • The third term is \[49 = 7^2\], a perfect square.
  • The middle term is -14x, which connects both perfect squares as \[2(7)(x)\].
This fits the form \[a^2 - 2ab + b^2\], indicating a perfect square trinomial. We can factor it as \[(x - 7)^2\].
Identifying these patterns is crucial for simplifying and solving quadratics efficiently.
Solving Quadratic Equations
Solving quadratic equations is a core skill in algebra. Once you have factored an equation, you can solve for the variable. For a factored form like \[(x - 7)^2 = 0\]:
  • Set the factor equal to zero: \[(x - 7) = 0\].
  • Solve for \[x\]: add 7 to both sides. Thus, \[x = 7\].
These steps are:
  1. Factor the quadratic equation.
  2. Set each factor equal to zero.
  3. Solve each resulting equation.
Consider another example like \[x^2 - 25 = 0\]:
  • Factor using the difference of squares: \[(x + 5)(x - 5) = 0\].
  • Set each factor equal to zero: \[(x + 5) = 0\] and \[(x - 5) = 0\].
  • Solve for \[x\]: \[x = -5\] and \[x = 5\].
Understanding these techniques simplifies solving quadratic equations, allowing us to find solutions quickly and efficiently.

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Most popular questions from this chapter

Expand each expression. $$ 9 c(-8+7 c) $$

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