Answer:

Quadratic equations

1. x^2 + 4x - 12 = 0

2. 3x^2 - 12x -15 = 0

3. 2x^2 + 2x - 4 = 0

4. x^2 - x - 6 = 0

Roots

1. x=2 and x=-6

2. x=5 and x=-1

3. x=1 and x=-2

4. x=3 and x=-2

Answer:

Here are the transformations and solutions for each equation:

1. x(x + 4) = 12

Expanding the equation, we get:

x^2 + 4x = 12

Rearranging the equation, we have:

x^2 + 4x - 12 = 0

Now we can solve this quadratic equation. Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 1, b = 4, and c = -12.

Plugging in these values, we have:

x = (-4 ± √(4^2 - 4(1)(-12))) / (2(1))

x = (-4 ± √(16 + 48)) / 2

x = (-4 ± √64) / 2

x = (-4 ± 8) / 2

This gives us two possible solutions:

x1 = (-4 + 8) / 2 = 4/2 = 2

x2 = (-4 - 8) / 2 = -12/2 = -6

Therefore, the roots of the equation are x = 2 and x = -6.

2. 3x(x - 4) = 15

Expanding the equation, we get:

3x^2 - 12x = 15

Rearranging the equation, we have:

3x^2 - 12x - 15 = 0

Now we can solve this quadratic equation. Dividing the equation by 3 to simplify, we get:

x^2 - 4x - 5 = 0

Using the quadratic formula, where a = 1, b = -4, and c = -5:

x = (-(-4) ± √((-4)^2 - 4(1)(-5))) / (2(1))

x = (4 ± √(16 + 20)) / 2

x = (4 ± √36) / 2

x = (4 ± 6) / 2

This gives us two possible solutions:

x1 = (4 + 6) / 2 = 10/2 = 5

x2 = (4 - 6) / 2 = -2/2 = -1

Therefore, the roots of the equation are x = 5 and x = -1.

3. 2x(1 + x) = 4

Expanding the equation, we get:

2x^2 + 2x = 4

Rearranging the equation, we have:

2x^2 + 2x - 4 = 0

Now we can solve this quadratic equation. Dividing the equation by 2 to simplify, we get:

x^2 + x - 2 = 0

Using the quadratic formula, where a = 1, b = 1, and c = -2:

x = (-1 ± √(1^2 - 4(1)(-2))) / (2(1))

x = (-1 ± √(1 + 8)) / 2

x = (-1 ± √9