Question
And hence, \( (x+\quad)^{2}= \) From here let's rearrange algebraically: (3) a) Take the square roots of both sides, don't forget to indicate there are two answers \( \qquad \) b) Simplify the fraction by recalling \( \sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}} \) \( \qquad \) Copyright reserved Please turn over Grade 11 Mathematics/Investigation 8 Limpopo DoE/Term 12025 c) Isolate the \( x \) on the left by moving the other term to the right \( \qquad \) d) Write the right-hand side as a single fraction
Ask by Weaver Burton. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
To solve \( (x + a)^2 = b \):
1. Take square roots: \( x + a = \pm \sqrt{b} \).
2. Isolate \( x \): \( x = -a \pm \sqrt{b} \).
3. Express as a single fraction: \( x = \frac{-a \pm \sqrt{b}}{1} \).
So, \( x = -a \pm \sqrt{b} \).
Solution
Let's solve the problem step by step, starting with the expression given:
### Step 1: Start with the expression
We have the expression \( (x + a)^2 = b \), where \( a \) and \( b \) are constants that we will need to define later.
### Step 2: Take the square roots of both sides
To take the square roots of both sides, we apply the square root to both sides of the equation. Remember that taking the square root introduces both a positive and a negative solution.
\[
\sqrt{(x + a)^2} = \pm \sqrt{b}
\]
This simplifies to:
\[
x + a = \pm \sqrt{b}
\]
### Step 3: Isolate \( x \)
Next, we isolate \( x \) by moving \( a \) to the right side of the equation:
\[
x = -a \pm \sqrt{b}
\]
### Step 4: Write the right-hand side as a single fraction
To express the right-hand side as a single fraction, we can rewrite it as follows:
\[
x = \frac{-a \pm \sqrt{b}}{1}
\]
This is already in the form of a single fraction, but if we want to express it in a more general form, we can write:
\[
x = \frac{-a + \sqrt{b}}{1} \quad \text{and} \quad x = \frac{-a - \sqrt{b}}{1}
\]
### Summary of the Steps
a) The square roots of both sides give us \( x + a = \pm \sqrt{b} \).
b) The simplification of the fraction is already done in the context of isolating \( x \).
c) We isolated \( x \) to get \( x = -a \pm \sqrt{b} \).
d) The right-hand side can be expressed as a single fraction, \( x = \frac{-a \pm \sqrt{b}}{1} \).
This concludes the algebraic rearrangement of the given expression. If you have specific values for \( a \) and \( b \), we can substitute them in for further simplification.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
A classic example of squaring a binomial is the expansion of \((x + a)^2\), which gives \(x^2 + 2ax + a^2\). This highlights how squaring affects both the variable and the constant. It’s a fundamental concept in algebra, helping us understand more complex equations and functions! In practical situations, knowing how to manipulate equations and isolate variables can come in handy! For example, if you're trying to determine how much of a specific ingredient you need in a recipe, you may need to rearrange quantities and solve for unknowns. Thus, mastering these algebraic fundamentals can streamline decision-making in everyday scenarios!
