Check whether the following are quadratic equations :
y(8y + 5) = `y^(2) + 3`

Answer

To determine whether the equation \( y(8y + 5) = y^2 + 3 \) is a quadratic equation, we will follow these steps:

### Step 1: Expand the left side of the equation
We start with the equation:
\[ y(8y + 5) = y^2 + 3 \]
Now, we will expand the left-hand side:
\[ 8y^2 + 5y = y^2 + 3 \]

### Step 2: Rearrange the equation
Next, we will move all terms to one side of the equation to set it to zero:
\[ 8y^2 + 5y - y^2 - 3 = 0 \]

### Step 3: Combine like terms
Now, we combine the like terms:
\[ (8y^2 - y^2) + 5y - 3 = 0 \]
This simplifies to:
\[ 7y^2 + 5y - 3 = 0 \]

### Step 4: Identify the coefficients
Now, we identify the coefficients \( a \), \( b \), and \( c \) from the standard form of a quadratic equation \( ay^2 + by + c = 0 \):
- \( a = 7 \)
- \( b = 5 \)
- \( c = -3 \)

### Step 5: Conclusion
Since the equation can be expressed in the form \( ay^2 + by + c = 0 \) where \( a \neq 0 \), we conclude that:
\[ 7y^2 + 5y - 3 = 0 \]
is indeed a quadratic equation.

### Final Answer
Yes, the given equation is a quadratic equation.

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