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) |
|=0This simplifies to:
|| x−2 | x−1 | x−1 |
| x−4 | x−3 | x−5 |
| x−8 | x−9 | x−25 |
|=0Next, 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 |
|=0Now, apply row operations:
Add the third row to the first row:
R1⟶R1+R3Add the third row to the second row:
R2⟶R2+R3This results in:
|| 0 | 2x−10 | 2x−26 |
| 0 | 2x−12 | 2x−30 |
| 1 | x−9 | x−25 |
|=0To solve, expand the determinant by the first column:
(2x−30)(2x−10)−(2x−12)(2x−26)=0This simplifies to:
2(x−15)⋅2(x−5)−2(x−6)⋅2(x−13)=0Further simplifying, we have:
(x−15)(x−5)−(x−6)(x−13)=0Solving this, we calculate:
x2−20x+75−(x2−19x+78)=0This leads to:
−x−3=0⟹x=−3Therefore,
x=−3 satisfies the equation
x2+2x−3=0.