Question

Directions: Solve using the square roots method. Write all answers in simplest form. Don't forget to show your work.

14. (x+4)^2 = -90

16. -(x-5)^2 = 108

18. 5(x-3)^2 = -225

20. -5/2(x+1)^2 = 30

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Answer 1

14. The solutions for
`\( (x+4)^2 = -90 \)` are complex:
`\( x = -4 + 6i \)` and
`\( x = -4 - 6i \)`.

16. The solutions for
`\( -(x-5)^2 = 108 \)` are complex:
`\( x = 5 + 6√(3)i \) and \( x = 5 - 6√(3)i \).`

18. The solutions for
`\( 5(x-3)^2 = -225 \)` are complex:
`\( x = 3 + 6i√(5) \)` and
`\( x = 3 - 6i√(5) \)`.

20. The solutions for
`\( -(5)/(2)(x+1)^2 = 30 \)` are complex:
`\( x = -1 + 2i√(3) \)` and
`\( x = -1 - 2i√(3) \)`.

Let's solve each equation step by step using the square roots method:

### 14.
`\( (x+4)^2 = -90 \)`

1. Expand the equation:

\((x+4)^2 = x^2 + 8x + 16\)

2. Set the equation equal to -90:

`\(x^2 + 8x + 16 = -90\)`

3. Move all terms to one side to form a quadratic equation:

`\(x^2 + 8x + 16 + 90 = 0\)`

4. Combine like terms:

`\(x^2 + 8x + 106 = 0\)`

5. Use the quadratic formula to find solutions:

`\[ x = (-b \pm √(b^2 - 4ac))/(2a) \]`

For this equation,
`\(a = 1\), \(b = 8\), and \(c = 106\).`

`\[ x = (-8 \pm √(8^2 - 4(1)(106)))/(2(1)) \]`

`\[ x = (-8 \pm √(64 - 424))/(2) \]`

`\[ x = (-8 \pm √(-360))/(2) \]`

The discriminant
`(\(b^2 - 4ac\))` is negative, so the solutions are complex. The solutions are:

`\[ x = -4 + 6i \quad \text{and} \quad x = -4 - 6i \]`

### 16.
`\( -(x-5)^2 = 108 \)`

1. Remove the negative sign by multiplying both sides by -1:

`\((x-5)^2 = -108\)`

2. Expand the equation:

`\((x-5)^2 = x^2 - 10x + 25\)`

3. Set the equation equal to -108:

`\(x^2 - 10x + 25 = -108\)`

4. Move all terms to one side:

`\(x^2 - 10x + 25 + 108 = 0\)`

5. Combine like terms:

`\(x^2 - 10x + 133 = 0\)`

6. Use the quadratic formula:

For this equation,
`\(a = 1\), \(b = -10\), and \(c = 133\).`

`\[ x = (10 \pm √((-10)^2 - 4(1)(133)))/(2(1)) \]`

`\[ x = (10 \pm √(100 - 532))/(2) \]`

`\[ x = (10 \pm √(-432))/(2) \]`

The discriminant is negative, so the solutions are complex. The solutions are:

`\[ x = 5 + 6√(3)i \quad \text{and} \quad x = 5 - 6√(3)i \]`

### 18.
`\( 5(x-3)^2 = -225 \)`

1. Divide both sides by 5 to simplify:

`\((x-3)^2 = -45\)`

2. Expand the equation:

`\((x-3)^2 = x^2 - 6x + 9\)`

3. Set the equation equal to -45:

`\(x^2 - 6x + 9 = -45\)`

4. Move all terms to one side:

`\(x^2 - 6x + 9 + 45 = 0\)`

5. Combine like terms:

`\(x^2 - 6x + 54 = 0\)`

6. Use the quadratic formula:

For this equation,
`\(a = 1\), \(b = -6\), and \(c = 54\)`.

`\[ x = (6 \pm √((-6)^2 - 4(1)(54)))/(2(1)) \]`

`\[ x = (6 \pm √(36 - 216))/(2) \]`

`\[ x = (6 \pm √(-180))/(2) \]`

The discriminant is negative, so the solutions are complex. The solutions are:

`\[ x = 3 + 6i√(5) \quad \text{and} \quad x = 3 - 6i√(5) \]`

### 20.
`\( -(5)/(2)(x+1)^2 = 30 \)`

1. Divide both sides by
`\(-(5)/(2)\)` to simplify:

`\((x+1)^2 = -12\)`

2. Expand the equation:

`\((x+1)^2 = x^2 + 2x + 1\)`

3. Set the equation equal to -12:

`\(x^2 + 2x + 1 = -12\)`

4. Move all terms to one side:

`\(x^2 + 2x + 1 + 12 = 0\)`

5. Combine like terms:

`\(x^2 + 2x + 13 = 0\)`

6. Use the quadratic formula:

For this equation,
`\(a = 1\), \(b = 2\), and \(c = 13\).`

`\[ x = (-2 \pm √(2^2 - 4(1)(13)))/(2(1)) \]`

`\[ x = (-2 \pm √(4 - 52))/(2) \]`

`\[ x = (-2 \pm √(-48))/(2) \]`

The discriminant is negative, so the solutions are complex. The solutions are:

`\[ x = -1 + 2i√(3) \quad \text{and} \quad x = -1 - 2i√(3) \]`

These are the solutions for the given equations, expressed in complex form.