To find the zeroes of the polynomial [tex]\( x^2 + 2\sqrt{2}x - 6 \)[/tex] and verify the relationships between the zeroes and the coefficients, follow these steps:
### Step 1: Identify the coefficients
The given polynomial is [tex]\( x^2 + 2\sqrt{2}x - 6 \)[/tex]. Here, the coefficients are:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 2\sqrt{2} \][/tex]
[tex]\[ c = -6 \][/tex]
### Step 2: Use the quadratic formula
To find the zeroes [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] of the polynomial, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ x = \frac{-(2\sqrt{2}) \pm \sqrt{(2\sqrt{2})^2 - 4(1)(-6)}}{2(1)} \][/tex]
[tex]\[ x = \frac{-2\sqrt{2} \pm \sqrt{8 + 24}}{2} \][/tex]
[tex]\[ x = \frac{-2\sqrt{2} \pm \sqrt{32}}{2} \][/tex]
[tex]\[ x = \frac{-2\sqrt{2} \pm 4\sqrt{2}}{2} \][/tex]
### Step 3: Simplify the expressions
Simplify the expressions to find the zeroes:
[tex]\[ x_1 = \frac{-2\sqrt{2} + 4\sqrt{2}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2} \][/tex]
[tex]\[ x_2 = \frac{-2\sqrt{2} - 4\sqrt{2}}{2} = \frac{-6\sqrt{2}}{2} = -3\sqrt{2} \][/tex]
So, the zeroes of the polynomial are:
[tex]\[ x_1 \approx 1.4142135623730951 \][/tex]
[tex]\[ x_2 \approx -4.242640687119285 \][/tex]
### Step 4: Verify the sum and product of the zeroes
According to the relationships between the zeroes and the coefficients of the polynomial [tex]\( ax^2 + bx + c \)[/tex]:
- The sum of the zeroes [tex]\( x_1 + x_2 \)[/tex] should be [tex]\( -\frac{b}{a} \)[/tex].
- The product of the zeroes [tex]\( x_1 \cdot x_2 \)[/tex] should be [tex]\( \frac{c}{a} \)[/tex].
#### Sum of the zeroes:
[tex]\[ x_1 + x_2 = \sqrt{2} + (-3\sqrt{2}) = -2\sqrt{2} \][/tex]
[tex]\[ \text{Sum of zeroes} = -\frac{b}{a} = -\frac{2\sqrt{2}}{1} = -2\sqrt{2} \approx -2.8284271247461903 \][/tex]
#### Product of the zeroes:
[tex]\[ x_1 \cdot x_2 = \sqrt{2} \cdot (-3\sqrt{2}) = -6 \][/tex]
[tex]\[ \text{Product of zeroes} = \frac{c}{a} = \frac{-6}{1} = -6 \][/tex]
### Conclusion
The zeroes of the polynomial [tex]\( x^2 + 2\sqrt{2}x - 6 \)[/tex] are approximately [tex]\( 1.4142135623730951 \)[/tex] and [tex]\( -4.242640687119285 \)[/tex]. The sum of the zeroes is [tex]\( -2.8284271247461903 \)[/tex], and the product of the zeroes is [tex]\( -6 \)[/tex]. These results confirm the relationships between the zeroes and the coefficients of the polynomial.
