Fill in the blanks:
(?×x−1)2+(?×x−2)2=5x2−8x+5
To solve this problem, we'll fill in the blanks in the expression (?×x−1)2+(?×x−2)2=5x2−8x+5.
Let's expand the individual expressions:
1. (a⋅x−1)2=a2x2−2ax+1
2. (b⋅x−2)2=b2x2−4bx+4
Now, add these expansions together:
(a2x2−2ax+1)+(b2x2−4bx+4)
= (a2+b2)x2−(2a+4b)x+5
We equate this to the given quadratic: 5x2−8x+5.
Matching coefficients, we have:
a2+b2=5
2a+4b=8
Now, solve these equations:
From the second equation: 2a+4b=8, divide by 2:
a+2b=4
Substitute a=4−2b from this into a2+b2=5:
(4−2b)2+b2=5
16−16b+4b2+b2=5
5b2−16b+16=5
5b2−16b+11=0
Factoring the quadratic equation 5b2−16b+11=0:
(5b−11)(b−1)=0
This gives us b=511 or b=1.
Using b=1, substitute back to find a:
a=4−2(1)=2
Thus, the correct integers for the blanks are a=2 and b=1.
Therefore, the correct answer is 2, 1.