Set A is having roots of equation `x^2+3x-10=0' .The set `A` is equal to (a) `[-5,-2]` (b) `[2,5]` (c) `[-5,2]` (d) `[-3,-2]`

Answer

To find the roots of the equation \( x^2 + 3x - 10 = 0 \) and determine the set \( A \), we will follow these steps:

### Step 1: Write down the quadratic equation
The given quadratic equation is:
\[ x^2 + 3x - 10 = 0 \]

### Step 2: Factor the quadratic equation
To factor the quadratic equation, we need to find two numbers that multiply to \(-10\) (the constant term) and add up to \(3\) (the coefficient of \(x\)). The numbers that satisfy this are \(5\) and \(-2\).

Thus, we can rewrite the equation as:
\[ x^2 + 5x - 2x - 10 = 0 \]
Grouping the terms:
\[ (x^2 + 5x) + (-2x - 10) = 0 \]
Factoring by grouping:
\[ x(x + 5) - 2(x + 5) = 0 \]
This can be factored as:
\[ (x - 2)(x + 5) = 0 \]

### Step 3: Solve for the roots
Setting each factor equal to zero gives us the roots:
1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x + 5 = 0 \) → \( x = -5 \)

Thus, the roots of the equation are \( x = 2 \) and \( x = -5 \).

### Step 4: Form the set A
The set \( A \) containing the roots is:
\[ A = \{-5, 2\} \]

### Step 5: Identify the correct option
Now, we compare this set with the given options:
(a) \([-5, -2]\)
(b) \([2, 5]\)
(c) \([-5, 2]\)
(d) \([-3, -2]\)

The correct representation of set \( A \) is option (c) \([-5, 2]\).

### Final Answer:
The set \( A \) is equal to \([-5, 2]\).

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