To determine the area of the rectangle whose dimensions are the roots of the quadratic equation [tex]\( x^2 - kx + 10 = 0 \)[/tex], we proceed as follows:
1. Identify the quadratic equation and its roots:
The given quadratic equation is [tex]\( x^2 - kx + 10 = 0 \)[/tex].
2. Find the roots:
Let's denote the roots of the quadratic equation by [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]. For any quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], the sum and product of the roots can be found using the relationships:
[tex]\[
\alpha + \beta = -\frac{b}{a} \quad \text{and} \quad \alpha \beta = \frac{c}{a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -k \)[/tex], and [tex]\( c = 10 \)[/tex]. Thus:
[tex]\[
\alpha + \beta = -\left(-\frac{k}{1}\right) = k
\][/tex]
[tex]\[
\alpha \beta = \frac{10}{1} = 10
\][/tex]
3. Calculate the area of the rectangle:
The area of the rectangle with dimensions [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] is given by the product of its roots:
[tex]\[
\text{Area} = \alpha \beta
\][/tex]
From the product of the roots we found:
[tex]\[
\alpha \beta = 10
\][/tex]
4. Select the correct option:
The area of the rectangle is [tex]\( 10 \)[/tex].
Therefore, the correct answer is:
1) [tex]\( 10 \)[/tex]
