Check whether the following are quadratic equations :
`y(2y + 15) = 2(y^(2) + y + 8)`

Answer

To determine whether the given equation \( y(2y + 15) = 2(y^2 + y + 8) \) is a quadratic equation, we will simplify and rearrange it step by step.

### Step 1: Expand both sides of the equation
We start by expanding both sides of the equation.

\[ y(2y + 15) = 2(y^2 + y + 8) \]

Expanding the left side:
\[ 2y^2 + 15y \]

Expanding the right side:
\[ 2y^2 + 2y + 16 \]

So, we rewrite the equation as:
\[ 2y^2 + 15y = 2y^2 + 2y + 16 \]

### Step 2: Move all terms to one side
Next, we will move all terms to one side of the equation to set it to zero.

\[ 2y^2 + 15y - 2y^2 - 2y - 16 = 0 \]

### Step 3: Combine like terms
Now, we combine the like terms:

\[ (2y^2 - 2y^2) + (15y - 2y) - 16 = 0 \]

This simplifies to:
\[ 13y - 16 = 0 \]

### Step 4: Identify the form of the equation
The equation we have now is:
\[ 13y - 16 = 0 \]

This can be rewritten as:
\[ 13y + 0y^2 - 16 = 0 \]

### Step 5: Check if it is a quadratic equation
A quadratic equation is generally in the form of:
\[ ax^2 + bx + c = 0 \]
where \( a \neq 0 \).

In our case:
- \( a = 0 \) (since there is no \( y^2 \) term),
- \( b = 13 \),
- \( c = -16 \).

Since \( a = 0 \), this equation is not a quadratic equation.

### Conclusion
Thus, the given equation \( y(2y + 15) = 2(y^2 + y + 8) \) is **not a quadratic equation**.

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