If `alphaandbeta` are the roots of a quadratic equation such that `alpha+beta=24andalpha-beta=8` , then the equation is

Options

A. `x^(2)-24x-128=0`
B. `x^(2)+24x+128=0`
C. `x^(2)+24x-128=0`
D. None of these

Answer

To find the quadratic equation given that the roots \( \alpha \) and \( \beta \) satisfy the conditions \( \alpha + \beta = 24 \) and \( \alpha - \beta = 8 \), we can follow these steps:

### Step 1: Find the values of \( \alpha \) and \( \beta \)

We have two equations:
1. \( \alpha + \beta = 24 \) (Equation 1)
2. \( \alpha - \beta = 8 \) (Equation 2)

To find \( \alpha \) and \( \beta \), we can add these two equations.

\[ (\alpha + \beta) + (\alpha - \beta) = 24 + 8 \]

This simplifies to:

\[ 2\alpha = 32 \]

Now, divide both sides by 2:

\[ \alpha = 16 \]

### Step 2: Substitute \( \alpha \) back to find \( \beta \)

Now that we have \( \alpha \), we can substitute it back into Equation 1 to find \( \beta \):

\[ 16 + \beta = 24 \]

Subtract 16 from both sides:

\[ \beta = 24 - 16 = 8 \]

### Step 3: Write the quadratic equation

The standard form of a quadratic equation with roots \( \alpha \) and \( \beta \) is given by:

\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]

We already know \( \alpha + \beta = 24 \) and now we need to calculate \( \alpha \beta \):

\[ \alpha \beta = 16 \times 8 = 128 \]

Now we can substitute these values into the quadratic equation:

\[ x^2 - 24x + 128 = 0 \]

### Final Answer

Thus, the quadratic equation is:

\[ x^2 - 24x + 128 = 0 \]

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