Answer :
Sure! Let's convert each of the given equations into the form [tex]\((x-p)^2 = q\)[/tex] by completing the square. I'll take each one step by step:
### Equation 24: [tex]\(0 = x^2 - 18x + 64\)[/tex]
1. Rearrange the equation: [tex]\(x^2 - 18x + 64 = 0\)[/tex].
2. Focus on the expression for completing the square: [tex]\(x^2 - 18x\)[/tex].
3. Find the value to complete the square: [tex]\((-18/2)^2 = 81\)[/tex].
4. Rewrite the equation: [tex]\((x^2 - 18x + 81) - 81 + 64 = 0\)[/tex].
5. Simplify: [tex]\((x - 9)^2 - 17 = 0\)[/tex].
6. Final form: [tex]\((x - 9)^2 = 17\)[/tex].
### Equation 25: [tex]\(x^2 + 22x + 120.5 = 0\)[/tex]
1. Rearrange the equation: [tex]\(x^2 + 22x + 120.5 = 0\)[/tex].
2. Focus on the expression for completing the square: [tex]\(x^2 + 22x\)[/tex].
3. Find the value to complete the square: [tex]\((22/2)^2 = 121\)[/tex].
4. Rewrite the equation: [tex]\((x^2 + 22x + 121) - 121 + 120.5 = 0\)[/tex].
5. Simplify: [tex]\((x + 11)^2 - 0.5 = 0\)[/tex].
6. Final form: [tex]\((x + 11)^2 = 0.5\)[/tex].
### Equation 26: [tex]\(x^2 + 3x - \frac{27}{4} = 0\)[/tex]
1. Rearrange the equation: [tex]\(x^2 + 3x - \frac{27}{4} = 0\)[/tex].
2. Focus on the expression for completing the square: [tex]\(x^2 + 3x\)[/tex].
3. Find the value to complete the square: [tex]\((3/2)^2 = \frac{9}{4}\)[/tex].
4. Rewrite the equation: [tex]\((x^2 + 3x + \frac{9}{4}) - \frac{9}{4} - \frac{27}{4} = 0\)[/tex].
5. Simplify: [tex]\((x + \frac{3}{2})^2 - \frac{36}{4} = 0\)[/tex].
6. Final form: [tex]\((x + \frac{3}{2})^2 = \frac{33}{4}\)[/tex].
### Equation 27: [tex]\(0 = 4x^2 + 4x - 14\)[/tex]
1. Rearrange the equation: [tex]\(4x^2 + 4x - 14 = 0\)[/tex].
2. Factor out the common term: [tex]\(4(x^2 + x) = 14\)[/tex].
3. Complete the square within the bracket: [tex]\((1/2)^2 = \frac{1}{4}\)[/tex].
4. Rewrite: [tex]\(4((x + \frac{1}{2})^2 - \frac{1}{4}) = 14\)[/tex].
5. Simplify: [tex]\((2x + 1)^2 - 1 = 14\)[/tex].
6. Final form: [tex]\((2x + 1)^2 = 15\)[/tex].
### Equation 28: [tex]\(0 = x^2 - \frac{3}{2}x - \frac{70}{8}\)[/tex]
1. Rearrange the equation: [tex]\(x^2 - \frac{3}{2}x - \frac{70}{8} = 0\)[/tex].
2. Focus on the expression: [tex]\(x^2 - \frac{3}{2}x\)[/tex].
3. Find the value to complete the square: [tex]\((-\frac{3}{4})^2 = \frac{9}{16}\)[/tex].
4. Rewrite the equation: [tex]\((x^2 - \frac{3}{2}x + \frac{9}{16}) - \frac{9}{16} - \frac{70}{8} = 0\)[/tex].
5. Simplify: [tex]\((x - \frac{3}{4})^2 - \frac{90}{16} = 0\)[/tex].
6. Final form: [tex]\((x - \frac{3}{4})^2 = \frac{90}{16}\)[/tex].
### Equation 29: [tex]\(x^2 + 0.6x - 19.1 = 1\)[/tex]
1. Rearrange the equation: [tex]\(x^2 + 0.6x - 19.1 = 1\)[/tex].
2. Move 1 to the left: [tex]\(x^2 + 0.6x - 20.1 = 0\)[/tex].
3. Focus on: [tex]\(x^2 + 0.6x\)[/tex].
4. Find the value to complete the square: [tex]\((0.3)^2 = 0.09\)[/tex].
5. Rewrite: [tex]\((x^2 + 0.6x + 0.09) - 0.09 - 20.1 = 0\)[/tex].
6. Simplify: [tex]\((x + 0.3)^2 - 20 = 0\)[/tex].
7. Final form: [tex]\((x + 0.3)^2 = 20\)[/tex].
I hope these steps help you understand how to complete the square for each equation!
### Equation 24: [tex]\(0 = x^2 - 18x + 64\)[/tex]
1. Rearrange the equation: [tex]\(x^2 - 18x + 64 = 0\)[/tex].
2. Focus on the expression for completing the square: [tex]\(x^2 - 18x\)[/tex].
3. Find the value to complete the square: [tex]\((-18/2)^2 = 81\)[/tex].
4. Rewrite the equation: [tex]\((x^2 - 18x + 81) - 81 + 64 = 0\)[/tex].
5. Simplify: [tex]\((x - 9)^2 - 17 = 0\)[/tex].
6. Final form: [tex]\((x - 9)^2 = 17\)[/tex].
### Equation 25: [tex]\(x^2 + 22x + 120.5 = 0\)[/tex]
1. Rearrange the equation: [tex]\(x^2 + 22x + 120.5 = 0\)[/tex].
2. Focus on the expression for completing the square: [tex]\(x^2 + 22x\)[/tex].
3. Find the value to complete the square: [tex]\((22/2)^2 = 121\)[/tex].
4. Rewrite the equation: [tex]\((x^2 + 22x + 121) - 121 + 120.5 = 0\)[/tex].
5. Simplify: [tex]\((x + 11)^2 - 0.5 = 0\)[/tex].
6. Final form: [tex]\((x + 11)^2 = 0.5\)[/tex].
### Equation 26: [tex]\(x^2 + 3x - \frac{27}{4} = 0\)[/tex]
1. Rearrange the equation: [tex]\(x^2 + 3x - \frac{27}{4} = 0\)[/tex].
2. Focus on the expression for completing the square: [tex]\(x^2 + 3x\)[/tex].
3. Find the value to complete the square: [tex]\((3/2)^2 = \frac{9}{4}\)[/tex].
4. Rewrite the equation: [tex]\((x^2 + 3x + \frac{9}{4}) - \frac{9}{4} - \frac{27}{4} = 0\)[/tex].
5. Simplify: [tex]\((x + \frac{3}{2})^2 - \frac{36}{4} = 0\)[/tex].
6. Final form: [tex]\((x + \frac{3}{2})^2 = \frac{33}{4}\)[/tex].
### Equation 27: [tex]\(0 = 4x^2 + 4x - 14\)[/tex]
1. Rearrange the equation: [tex]\(4x^2 + 4x - 14 = 0\)[/tex].
2. Factor out the common term: [tex]\(4(x^2 + x) = 14\)[/tex].
3. Complete the square within the bracket: [tex]\((1/2)^2 = \frac{1}{4}\)[/tex].
4. Rewrite: [tex]\(4((x + \frac{1}{2})^2 - \frac{1}{4}) = 14\)[/tex].
5. Simplify: [tex]\((2x + 1)^2 - 1 = 14\)[/tex].
6. Final form: [tex]\((2x + 1)^2 = 15\)[/tex].
### Equation 28: [tex]\(0 = x^2 - \frac{3}{2}x - \frac{70}{8}\)[/tex]
1. Rearrange the equation: [tex]\(x^2 - \frac{3}{2}x - \frac{70}{8} = 0\)[/tex].
2. Focus on the expression: [tex]\(x^2 - \frac{3}{2}x\)[/tex].
3. Find the value to complete the square: [tex]\((-\frac{3}{4})^2 = \frac{9}{16}\)[/tex].
4. Rewrite the equation: [tex]\((x^2 - \frac{3}{2}x + \frac{9}{16}) - \frac{9}{16} - \frac{70}{8} = 0\)[/tex].
5. Simplify: [tex]\((x - \frac{3}{4})^2 - \frac{90}{16} = 0\)[/tex].
6. Final form: [tex]\((x - \frac{3}{4})^2 = \frac{90}{16}\)[/tex].
### Equation 29: [tex]\(x^2 + 0.6x - 19.1 = 1\)[/tex]
1. Rearrange the equation: [tex]\(x^2 + 0.6x - 19.1 = 1\)[/tex].
2. Move 1 to the left: [tex]\(x^2 + 0.6x - 20.1 = 0\)[/tex].
3. Focus on: [tex]\(x^2 + 0.6x\)[/tex].
4. Find the value to complete the square: [tex]\((0.3)^2 = 0.09\)[/tex].
5. Rewrite: [tex]\((x^2 + 0.6x + 0.09) - 0.09 - 20.1 = 0\)[/tex].
6. Simplify: [tex]\((x + 0.3)^2 - 20 = 0\)[/tex].
7. Final form: [tex]\((x + 0.3)^2 = 20\)[/tex].
I hope these steps help you understand how to complete the square for each equation!
