To solve the given determinant equation, we start by simplifying the expression: |
x−2
3(x−1)
5(x−1)
x−4
3(x−3)
5(x−5)
x−8
3(x−9)
5(x−25)
|=0 This simplifies to: |
x−2
x−1
x−1
x−4
x−3
x−5
x−8
x−9
x−25
|=0 Next, perform column operations to simplify the determinant. Subtract the second column from the first: C1⟶C1−C2⟹|
−1
x−1
x−1
−1
x−3
x−5
1
x−9
x−25
|=0 Now, apply row operations: Add the third row to the first row: R1⟶R1+R3 Add the third row to the second row: R2⟶R2+R3 This results in: |
0
2x−10
2x−26
0
2x−12
2x−30
1
x−9
x−25
|=0
To solve, expand the determinant by the first column: (2x−30)(2x−10)−(2x−12)(2x−26)=0 This simplifies to: 2(x−15)⋅2(x−5)−2(x−6)⋅2(x−13)=0 Further simplifying, we have: (x−15)(x−5)−(x−6)(x−13)=0 Solving this, we calculate: x2−20x+75−(x2−19x+78)=0 This leads to: −x−3=0⟹x=−3 Therefore, x=−3 satisfies the equation x2+2x−3=0.