1 For each of the following equations: i find what must be added to both sides of the equation to create a perfect square on the LHS ii write each equation in the form \( (x+p)^{2}=k \) \( \begin{array}{lll}\text { a } x^{2}+2 x=5 & \text { b } x^{2}-2 x=-7 & \text { c } x^{2}+6 x=2 \\ \text { d } x^{2}-6 x=-3 & \text { e } x^{2}+10 x=1 & \text { f } x^{2}-8 x=5 \\ \text { g } x^{2}+12 x=13 & \text { h } x^{2}+5 x=-2 & \text { i } x^{2}-7 x=4\end{array} \)

**a) \( x^{2}+2x=5 \)** 1. Identify the term to complete the square. For \( x^2 + 2x \), the coefficient of \( x \) is \( 2 \). Half of \( 2 \) is \( 1 \) and its square is \[ \left(\frac{2}{2}\right)^2 = 1. \] 2. Add \( 1 \) to both sides: \[ x^2 + 2x + 1 = 5 + 1. \] 3. Write the left-hand side as a perfect square: \[ (x + 1)^2 = 6. \] --- **b) \( x^{2}-2x=-7 \)** 1. For \( x^2 - 2x \), the coefficient of \( x \) is \( -2 \). Half of \( -2 \) is \( -1 \) and its square is \[ \left(\frac{-2}{2}\right)^2 = 1. \] 2. Add \( 1 \) to both sides: \[ x^2 - 2x + 1 = -7 + 1. \] 3. Write the left-hand side as a perfect square: \[ (x - 1)^2 = -6. \] --- **c) \( x^{2}+6x=2 \)** 1. For \( x^2 + 6x \), the coefficient of \( x \) is \( 6 \). Half of \( 6 \) is \( 3 \) and its square is \[ \left(\frac{6}{2}\right)^2 = 9. \] 2. Add \( 9 \) to both sides: \[ x^2 + 6x + 9 = 2 + 9. \] 3. Write the left-hand side as a perfect square: \[ (x + 3)^2 = 11. \] --- **d) \( x^{2}-6x=-3 \)** 1. For \( x^2 - 6x \), the coefficient of \( x \) is \( -6 \). Half of \( -6 \) is \( -3 \) and its square is \[ \left(\frac{-6}{2}\right)^2 = 9. \] 2. Add \( 9 \) to both sides: \[ x^2 - 6x + 9 = -3 + 9. \] 3. Write the left-hand side as a perfect square: \[ (x - 3)^2 = 6. \] --- **e) \( x^{2}+10x=1 \)** 1. For \( x^2 + 10x \), the coefficient of \( x \) is \( 10 \). Half of \( 10 \) is \( 5 \) and its square is \[ \left(\frac{10}{2}\right)^2 = 25. \] 2. Add \( 25 \) to both sides: \[ x^2 + 10x + 25 = 1 + 25. \] 3. Write the left-hand side as a perfect square: \[ (x + 5)^2 = 26. \] --- **f) \( x^{2}-8x=5 \)** 1. For \( x^2 - 8x \), the coefficient of \( x \) is \( -8 \). Half of \( -8 \) is \( -4 \) and its square is \[ \left(\frac{-8}{2}\right)^2 = 16. \] 2. Add \( 16 \) to both sides: \[ x^2 - 8x + 16 = 5 + 16. \] 3. Write the left-hand side as a perfect square: \[ (x - 4)^2 = 21. \] --- **g) \( x^{2}+12x=13 \)** 1. For \( x^2 + 12x \), the coefficient of \( x \) is \( 12 \). Half of \( 12 \) is \( 6 \) and its square is \[ \left(\frac{12}{2}\right)^2 = 36. \] 2. Add \( 36 \) to both sides: \[ x^2 + 12x + 36 = 13 + 36. \] 3. Write the left-hand side as a perfect square: \[ (x + 6)^2 = 49. \] --- **h) \( x^{2}+5x=-2 \)** 1. For \( x^2 + 5x \), the coefficient of \( x \) is \( 5 \). Half of \( 5 \) is \( \frac{5}{2} \) and its square is \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4}. \] 2. Add \( \frac{25}{4} \) to both sides: \[ x^2 + 5x + \frac{25}{4} = -2 + \frac{25}{4}. \] 3. Simplify the right-hand side: \(-2 = -\frac{8}{4}\), so \[ -\frac{8}{4} + \frac{25}{4} = \frac{17}{4}. \] 4. Write the left-hand side as a perfect square: \[ \left(x + \frac{5}{2}\right)^2 = \frac{17}{4}. \] --- **i) \( x^{2}-7x=4 \)** 1. For \( x^2 - 7x \), the coefficient of \( x \) is \( -7 \). Half of \( -7 \) is \( -\frac{7}{2} \) and its square is \[ \left(\frac{-7}{2}\right)^2 = \frac{49}{4}. \] 2. Add \( \frac{49}{4} \) to both sides: \[ x^2 - 7x + \frac{49}{4} = 4 + \frac{49}{4}. \] 3. Simplify the right-hand side: \(4 = \frac{16}{4}\), so \[ \frac{16}{4} + \frac{49}{4} = \frac{65}{4}. \] 4. Write the left-hand side as a perfect square: \[ \left(x - \frac{7}{2}\right)^2 = \frac{65}{4}. \]