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Chapter 0: Problem 82

Solve the equation by extracting square roots. $$ (x+5)^{2}=(x+4)^{2} $$

Short Answer

Expert verified
The solution to the equation \((x+5)^{2} = (x+4)^{2}\) is \(x = - \frac{9}{2}\)

Step by step solution

01

Expand Both Sides of the Equation

Start off with expanding the perfect squares of both sides of the equation. This gives us:\n\n\[(x+5)^{2} = (x+4)^{2}\] becomes \[x^{2} + 10x + 25 = x^{2} + 8x + 16\]
02

Simplify the Equation

The next step is then to simplify the equation by removing like terms from both sides of the equation. Subtract \(x^{2}\) and \(8x\) from both sides of the equation. This results in:\n\n\[2x + 9 = 0.\]
03

Solve the Equation

Finally, isolate the variable 'x' by subtracting '9' from both sides and then dividing by '2'. This yields the value of x in the given equation.\n\nSolving this yields: \(x = - \frac{9}{2}\)

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Key Concepts

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

Factoring
Factoring is a fundamental technique used to solve quadratic equations by expressing the equation as a product of its factors. Imagine an equation like \( ax^2 + bx + c = 0 \). By factoring, we want to express this as \( (mx + n)(px + q) = 0 \).
  • This involves identifying two numbers whose product is equal to \( ac \), and whose sum is \( b \).
  • Once the correct pairs are found, they replace the middle term in the equation, facilitating further breaking down of the equation into two binomials.
  • This process simplifies finding the roots, because if \((mx + n)(px + q) = 0 \), then \(mx + n = 0\) or \(px + q = 0\).
This approach is especially practical for quadratic equations that can be easily decomposed, allowing you to quickly find the solution by solving simpler linear equations. It transforms an initially complex problem into a more manageable one.
Perfect Squares
Perfect squares are expressions like \((x + a)^2\) or \(x^2 + 2ax + a^2\) where a binomial is squared. Recognizing these can greatly simplify solving quadratic equations, especially when both sides of an equation are perfect squares.
  • Take \((x + 5)^2 = (x + 4)^2\). Both sides are perfectly squared, implying that their bases are equal if there's no other solution affecting it.
  • Expanding the squares results in \(x^2 + 10x + 25 = x^2 + 8x + 16\), showing that similarities in equations can be canceled out or simplified.
  • Once expanded, such terms help in revealing an underlying simpler equation or numerical terms that can lead to straightforward solutions.
Identifying and working with perfect squares streamlines the equation-solving process, often bypassing more intricate calculations and leading you directly to the solution.
Quadratic Formula
The quadratic formula is a universal method used for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It's expressed as:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • This formula derives from completing the square and is particularly useful when factoring is not apparent.
  • The term \(b^2 - 4ac\), called the discriminant, reveals the nature of the equation's roots: real and distinct, real and repeated, or non-real.
Utilizing the quadratic formula is advantageous as it provides a direct, surefire way to ascertain the roots of the quadratic equation, ensuring that no possible solutions are overlooked. While other methods may be easier for some problems, this formula is a reliable staple for any quadratic problem.

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Most popular questions from this chapter

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. Proof Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from a height of \(6.5\) feet at a velocity of 72 feet per second. $$ t_{1}=0, t_{2}=4 $$

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from ground level at a velocity of 120 feet per second. $$ t_{1}=3, t_{2}=5 $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=-\frac{2}{x} $$

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is dropped from a height of 80 feet. $$ t_{1}=1, t_{2}=2 $$
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