Certainly! Let's solve the given quadratic equation step by step.
### Given:
The quadratic equation is [tex]\(2x^2 + kx - 2 = 0\)[/tex] and one of its roots is [tex]\( x_1 = 2 \)[/tex].
### Part (a): Determine the value of [tex]\( k \)[/tex].
#### Step 1: Use Vieta’s formulas which relate the coefficients of a polynomial to sums and products of its roots.
For a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], Vieta's Formulas tell us:
1. The sum of the roots [tex]\((x_1 + x_2)\)[/tex] is equal to [tex]\( -\frac{b}{a} \)[/tex].
2. The product of the roots [tex]\((x_1 \cdot x_2)\)[/tex] is equal to [tex]\(\frac{c}{a}\)[/tex].
Here, [tex]\( a = 2 \)[/tex], [tex]\( b = k \)[/tex], and [tex]\( c = -2 \)[/tex].
#### Step 2: Apply Vieta’s formulas to find the sum of the roots.
Since one root is [tex]\( x_1 = 2 \)[/tex], let the other root be [tex]\( x_2 \)[/tex].
The sum of the roots is:
[tex]\[ x_1 + x_2 = -\frac{k}{2} \][/tex]
Substituting [tex]\( x_1 = 2 \)[/tex]:
[tex]\[ 2 + x_2 = -\frac{k}{2} \][/tex]
#### Step 3: Apply Vieta’s formulas to find the product of the roots.
The product of the roots is:
[tex]\[ x_1 \cdot x_2 = \frac{-2}{2} \][/tex]
[tex]\[ 2 \cdot x_2 = -1 \][/tex]
Solving for [tex]\( x_2 \)[/tex]:
[tex]\[ x_2 = -\frac{1}{2} \][/tex]
#### Step 4: Substitute [tex]\( x_2 \)[/tex] back into the sum of the roots equation to solve for [tex]\( k \)[/tex].
[tex]\[ 2 + \left(-\frac{1}{2}\right) = -\frac{k}{2} \][/tex]
[tex]\[ 2 - \frac{1}{2} = -\frac{k}{2} \][/tex]
[tex]\[ \frac{4}{2} - \frac{1}{2} = -\frac{k}{2} \][/tex]
[tex]\[ \frac{3}{2} = -\frac{k}{2} \][/tex]
Multiplying both sides by 2 to solve for [tex]\( k \)[/tex]:
[tex]\[ 3 = -k \][/tex]
Hence,
[tex]\[ k = -3 \][/tex]
### Part (b): What is the other root?
We have already determined the other root while solving for [tex]\( k \)[/tex].
From the product of the roots:
[tex]\[ x_1 \cdot x_2 = \frac{-2}{2} \][/tex]
[tex]\[ 2 \cdot x_2 = -1 \][/tex]
[tex]\[ x_2 = -\frac{1}{2} \][/tex]
### Final Answer
(a) The value of [tex]\( k \)[/tex] is [tex]\(-3\)[/tex].
(b) The other root is [tex]\(-\frac{1}{2}\)[/tex].
Summary:
1. [tex]\( k = -3 \)[/tex]
2. The other root [tex]\( x_2 = -\frac{1}{2} \)[/tex]
