What is the sum of the sqares of the roots of the equation `x^(2) + 2x - 143 = 0`
(a)170
(b)180
(c)190
(d)290

Options

A. 170
B. 180
C. 190
D. 290

Answer

To find the sum of the squares of the roots of the quadratic equation \( x^2 + 2x - 143 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = 2 \)
- \( c = -143 \)

### Step 2: Calculate the sum of the roots
Using Vieta's formulas, the sum of the roots \( \alpha + \beta \) can be calculated as:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{2}{1} = -2 \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{-143}{1} = -143 \]

### Step 4: Use the formula for the sum of the squares of the roots
The sum of the squares of the roots can be calculated using the formula:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (-2)^2 - 2(-143) \]

### Step 5: Calculate the values
Calculating the squares and products:
\[ \alpha^2 + \beta^2 = 4 + 2 \times 143 \]
\[ \alpha^2 + \beta^2 = 4 + 286 = 290 \]

### Final Answer
Thus, the sum of the squares of the roots is \( \boxed{290} \).

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