Problem 15
Question
Solve each quadratic equation by extraction of roots. $$ (x-5)^{2}=1 $$
Step-by-Step Solution
Verified Answer
Answer: The solutions to the given quadratic equation are \(x = 6\) and \(x = 4\).
1Step 1: Isolate the squared term
First, let's isolate the squared term \((x-5)^2\) by making sure it is on one side of the equation. In this case, it is already isolated:
$$
(x-5)^2 = 1
$$
2Step 2: Take the square root of both sides
Now, let's take the square root of both sides of the equation to eliminate the squared term. Remember that when taking the square root of a number, we must consider both the positive and negative solutions:
$$
\sqrt{(x-5)^2} = \pm\sqrt{1}
$$
3Step 3: Simplify the equation
Now that we have taken the square root of both sides, we can simplify the equation. The square root of \((x-5)^2\) is \(x-5\), and the square root of 1 is 1:
$$
x - 5 = \pm1
$$
4Step 4: Solve for x
Next, let's solve for x by considering both the positive and negative solutions.
1. Positive solution: When considering the positive solution (\(+1\)), we get:
$$
x - 5 = 1
$$
Add 5 to both sides:
$$
x = 1 + 5
$$
So, \(x = 6\).
2. Negative solution: When considering the negative solution (\(-1\)), we get:
$$
x - 5 = -1
$$
Add 5 to both sides:
$$
x = -1 + 5
$$
So, \(x = 4\).
5Step 5: State the final answer
Thus, the solutions to the given quadratic equation are \(x = 6\) and \(x = 4\).
Key Concepts
Extracting RootsSolving by Square RootsQuadratic Formula
Extracting Roots
Extracting roots is a method used to solve equations that involve squared terms. This is particularly useful in quadratic equations that are perfect squares, making it easier to find the solutions directly. In the exercise provided, we have the equation \((x-5)^2 = 1\). Here, our task is to "extract the root" to solve for \(x\).
This process involves two key steps: first, take the square root of both sides of the equation, which eliminates the square on the left-hand side. Secondly, remember that taking a square root results in two possible solutions, one positive and one negative, due to the nature of squaring a negative number (e.g., \( (-1)^2 = 1 \)). Thus, when we solve \( \sqrt{(x-5)^2} = \pm\sqrt{1} \), we must consider both \( +1 \) and \( -1 \) as potential outcomes. This step is crucial for completing the process of extracting roots properly.
In essence, extracting roots is about reversing the squaring process to find the original value, which helps in solving quadratic equations efficiently.
This process involves two key steps: first, take the square root of both sides of the equation, which eliminates the square on the left-hand side. Secondly, remember that taking a square root results in two possible solutions, one positive and one negative, due to the nature of squaring a negative number (e.g., \( (-1)^2 = 1 \)). Thus, when we solve \( \sqrt{(x-5)^2} = \pm\sqrt{1} \), we must consider both \( +1 \) and \( -1 \) as potential outcomes. This step is crucial for completing the process of extracting roots properly.
In essence, extracting roots is about reversing the squaring process to find the original value, which helps in solving quadratic equations efficiently.
Solving by Square Roots
Solving an equation by square roots is a straightforward method especially useful when dealing with quadratic equations in the form where the square term is isolated. From our example, \((x-5)^2 = 1\), we focus on solving it by square roots.
The process involves isolating the squared term, and in our given equation, this is already done. The next step is to apply square roots on both sides, indicated by \( \sqrt{(x-5)^2} = \pm\sqrt{1} \). It's important to remember that since we are applying square roots, our answer consists of two values, corresponding to both \(+1\) and \(-1\) from the right-hand side.
This gives us two simple linear equations to solve: \(x - 5 = +1\) and \(x - 5 = -1\). By solving these individually, you arrive at \(x = 6\) from the positive solution and \(x = 4\) from the negative solution.
The process involves isolating the squared term, and in our given equation, this is already done. The next step is to apply square roots on both sides, indicated by \( \sqrt{(x-5)^2} = \pm\sqrt{1} \). It's important to remember that since we are applying square roots, our answer consists of two values, corresponding to both \(+1\) and \(-1\) from the right-hand side.
This gives us two simple linear equations to solve: \(x - 5 = +1\) and \(x - 5 = -1\). By solving these individually, you arrive at \(x = 6\) from the positive solution and \(x = 4\) from the negative solution.
- Benefit: It is efficient and less complicated for certain quadratic forms.
- Limitation: It requires the quadratic equation to be in a particular form.
Quadratic Formula
The quadratic formula is another powerful tool for solving quadratic equations in the general form \(ax^2 + bx + c = 0\). For any quadratic equation, no matter how complex, the quadratic formula provides the solutions using: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The equation \((x-5)^2 = 1\) can also be expanded to \(x^2 - 10x + 25 = 1\) or \(x^2 - 10x + 24 = 0\) as a standard quadratic equation. In this context, the quadratic formula could be used by identifying \(a = 1\), \(b = -10\), and \(c = 24\).
However, using either extracting roots or solving by square roots is more efficient as the original equation is already a perfect square. But understanding the quadratic formula is vital as it ensures you have a reliable method that caters to all quadratic problems, especially those not simplified by prior methods.
The equation \((x-5)^2 = 1\) can also be expanded to \(x^2 - 10x + 25 = 1\) or \(x^2 - 10x + 24 = 0\) as a standard quadratic equation. In this context, the quadratic formula could be used by identifying \(a = 1\), \(b = -10\), and \(c = 24\).
However, using either extracting roots or solving by square roots is more efficient as the original equation is already a perfect square. But understanding the quadratic formula is vital as it ensures you have a reliable method that caters to all quadratic problems, especially those not simplified by prior methods.
- Pros: Universally applies to any quadratic equation.
- Cons: More compute-intensive compared to specific methods like extracting roots.
Other exercises in this chapter
Problem 15
For the following problems, solve the equations. $$ a^{2}+6 a+8=0 $$
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For the following problems, graph the quadratic equations. \(y=-x^{2}+1\) (Compare with problem 2.)
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For the following problems, solve the equations using the quadratic formula. $$ x^{2}-2 x-3=0 $$
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