If a,b are two numbers such that `a ^(2) +b^(2) =7 and a ^(3) + b^(3) =10,` then :

Options

A. The greatest value of `|a+b|=5`
B. The greatest value of `(a+b) ` is 4
C. The leatest value of `|a+b|` is 1
D. The least vlaue of `|a+b|` is 1

Answer

To solve the problem, we need to find the values of \( |a + b| \) given the equations:

1. \( a^2 + b^2 = 7 \)
2. \( a^3 + b^3 = 10 \)

We will use the identities and relationships between the sums and products of \( a \) and \( b \).

### Step 1: Use the identity for \( a^3 + b^3 \)

We know that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
From the first equation, we have \( a^2 + b^2 = 7 \). We can express \( a^3 + b^3 \) as:
\[ a^3 + b^3 = (a + b)((a^2 + b^2) - ab) = (a + b)(7 - ab) \]
Setting this equal to 10 gives us:
\[ (a + b)(7 - ab) = 10 \tag{1} \]

### Step 2: Express \( ab \) in terms of \( a + b \)

We also know that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting \( a^2 + b^2 = 7 \) into this equation, we get:
\[ 7 = (a + b)^2 - 2ab \]
Rearranging gives:
\[ 2ab = (a + b)^2 - 7 \implies ab = \frac{(a + b)^2 - 7}{2} \tag{2} \]

### Step 3: Substitute \( ab \) into equation (1)

Let \( x = a + b \). Then from equation (2), we can substitute \( ab \) into equation (1):
\[ x(7 - \frac{x^2 - 7}{2}) = 10 \]
Multiplying through by 2 to eliminate the fraction:
\[ 2x(7 - \frac{x^2 - 7}{2}) = 20 \]
This simplifies to:
\[ 2x(7) - x(x^2 - 7) = 20 \]
\[ 14x - x^3 + 7x = 20 \]
Combining like terms:
\[ 21x - x^3 = 20 \]
Rearranging gives us the cubic equation:
\[ x^3 - 21x + 20 = 0 \tag{3} \]

### Step 4: Solve the cubic equation

To find the roots of the cubic equation \( x^3 - 21x + 20 = 0 \), we can use the Rational Root Theorem or simply test possible rational roots. Testing \( x = 1 \):
\[ 1^3 - 21(1) + 20 = 1 - 21 + 20 = 0 \]
Thus, \( x = 1 \) is a root. We can factor the cubic polynomial as:
\[ (x - 1)(x^2 + x - 20) = 0 \]

Next, we solve the quadratic \( x^2 + x - 20 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1 + 80}}{2} = \frac{-1 \pm 9}{2} \]
This gives us:
\[ x = \frac{8}{2} = 4 \quad \text{and} \quad x = \frac{-10}{2} = -5 \]

### Step 5: Possible values of \( a + b \)

The possible values of \( a + b \) are \( 1, 4, -5 \).

### Step 6: Find the greatest and least values of \( |a + b| \)

Calculating the absolute values:
- \( |1| = 1 \)
- \( |4| = 4 \)
- \( |-5| = 5 \)

Thus, the greatest value of \( |a + b| \) is \( 5 \) and the least value is \( 1 \).

### Final Answer

- The greatest value of \( |a + b| \) is \( 5 \).
- The least value of \( |a + b| \) is \( 1 \).