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

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.