Of the following quadratic equations, which is the one whose roots are 2 and -15?

Options

A. `x^(2)-2x+15=0`
B. `x^(2)+15x-2=0`
C. `x^(2)+13x-30=0`
D. `x^(2)-30=0`

Answer

To find the quadratic equation whose roots are 2 and -15, we can follow these steps:

### Step 1: Understand the relationship between roots and factors
If a quadratic equation has roots \( \alpha \) and \( \beta \), then it can be expressed in factored form as:
\[ (x - \alpha)(x - \beta) = 0 \]

### Step 2: Identify the roots
In this case, the roots are given as:
\[ \alpha = 2 \quad \text{and} \quad \beta = -15 \]

### Step 3: Write the factors using the roots
Using the roots, we can write the factors:
\[ (x - 2) \quad \text{and} \quad (x + 15) \]
(Note that \( x - (-15) \) simplifies to \( x + 15 \))

### Step 4: Form the quadratic equation
Now, we can multiply the factors to form the quadratic equation:
\[ (x - 2)(x + 15) \]

### Step 5: Expand the equation
We will expand the equation:
\[ x^2 + 15x - 2x - 30 = 0 \]
Combining like terms gives:
\[ x^2 + 13x - 30 = 0 \]

### Step 6: Final quadratic equation
Thus, the quadratic equation whose roots are 2 and -15 is:
\[ x^2 + 13x - 30 = 0 \]

### Step 7: Check for options
Now, we can check the available options to see which one matches our derived equation.