What are the roots of the equations `2(y + 2)^(2) - 5(y + 2) = 12`?

Options

A. `-7//2.2`
B. `-3//2.4`
C. `-5//3.3`
D. `3//2.4`

Answer

To find the roots of the equation \(2(y + 2)^2 - 5(y + 2) = 12\), we can follow these steps:

### Step 1: Simplify the Equation
Start by rewriting the equation:
\[ 2(y + 2)^2 - 5(y + 2) - 12 = 0 \]

### Step 2: Substitute
Let \(m = y + 2\). Then the equation becomes:
\[ 2m^2 - 5m - 12 = 0 \]

### Step 3: Identify Coefficients
In the quadratic equation \(2m^2 - 5m - 12 = 0\), identify the coefficients:
- \(a = 2\)
- \(b = -5\)
- \(c = -12\)

### Step 4: Use the Quadratic Formula
The quadratic formula is given by:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ m = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-12)}}{2 \cdot 2} \]

### Step 5: Calculate the Discriminant
Calculate \(b^2 - 4ac\):
\[ (-5)^2 = 25 \]
\[ 4 \cdot 2 \cdot (-12) = -96 \quad \text{(the negative sign makes this a positive when subtracted)} \]
So,
\[ b^2 - 4ac = 25 + 96 = 121 \]

### Step 6: Substitute Back into the Formula
Now substitute back into the quadratic formula:
\[ m = \frac{5 \pm \sqrt{121}}{4} \]
Since \(\sqrt{121} = 11\):
\[ m = \frac{5 \pm 11}{4} \]

### Step 7: Solve for \(m\)
This gives us two possible values for \(m\):
1. \(m = \frac{5 + 11}{4} = \frac{16}{4} = 4\)
2. \(m = \frac{5 - 11}{4} = \frac{-6}{4} = -\frac{3}{2}\)

### Step 8: Substitute Back to Find \(y\)
Recall that \(m = y + 2\). Now we can find \(y\):
1. For \(m = 4\):
\[ 4 = y + 2 \implies y = 4 - 2 = 2 \]
2. For \(m = -\frac{3}{2}\):
\[ -\frac{3}{2} = y + 2 \implies y = -\frac{3}{2} - 2 = -\frac{3}{2} - \frac{4}{2} = -\frac{7}{2} \]

### Final Roots
Thus, the roots of the equation \(2(y + 2)^2 - 5(y + 2) = 12\) are:
\[ y = 2 \quad \text{and} \quad y = -\frac{7}{2} \]

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