To solve for [tex]\( M \)[/tex] given the equations [tex]\( a + b = 6 \)[/tex] and [tex]\( ab = 10 \)[/tex], we will follow a step-by-step approach to identify the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], and then compute [tex]\( M \)[/tex].
### Step 1: Form the Quadratic Equation
Given the sums and products of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we can form a quadratic equation:
[tex]\[ x^2 - (a + b)x + ab = 0 \][/tex]
Substituting [tex]\( a + b = 6 \)[/tex] and [tex]\( ab = 10 \)[/tex], we get:
[tex]\[ x^2 - 6x + 10 = 0 \][/tex]
### Step 2: Calculate the Discriminant
The discriminant [tex]\( \Delta \)[/tex] of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the quadratic equation [tex]\( x^2 - 6x + 10 = 0 \)[/tex], we have [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 10 \)[/tex]. Thus,
[tex]\[ \Delta = (-6)^2 - 4 \cdot 1 \cdot 10 = 36 - 40 = -4 \][/tex]
### Step 3: Calculate the Roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since [tex]\( \Delta = -4 \)[/tex], we have:
[tex]\[ x = \frac{6 \pm \sqrt{-4}}{2} = \frac{6 \pm 2i}{2} = 3 \pm i \][/tex]
Thus, the roots are:
[tex]\[ a = 3 + i \][/tex]
[tex]\[ b = 3 - i \][/tex]
### Step 4: Calculate [tex]\( a^2 + b^2 + ab \)[/tex]
Next, we need to find [tex]\( a^2 + b^2 + ab - 1 \)[/tex]:
1. Calculate [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex]:
[tex]\[ a^2 = (3+i)^2 = 9 + 6i + i^2 = 9 + 6i - 1 = 8 + 6i \][/tex]
[tex]\[ b^2 = (3-i)^2 = 9 - 6i + i^2 = 9 - 6i - 1 = 8 - 6i \][/tex]
2. Calculate [tex]\( a \cdot b \)[/tex]:
[tex]\[ ab = (3+i)(3-i) = 9 - i^2 = 9 - (-1) = 9 + 1 = 10 \][/tex]
3. Sum up [tex]\( a^2 + b^2 + ab \)[/tex]:
[tex]\[ a^2 + b^2 + ab = (8 + 6i) + (8 - 6i) + 10 = 8 + 8 + 10 = 26 \][/tex]
### Step 5: Compute [tex]\( M \)[/tex]
Finally, calculate [tex]\( M \)[/tex]:
[tex]\[ M = \sqrt{a^2 + b^2 + ab - 1} = \sqrt{26 - 1} = \sqrt{25} = 5 \][/tex]
### Conclusion
The value of [tex]\( M \)[/tex] for the given system of equations is:
[tex]\[ M = 5 \][/tex]
