Find the Coefficients: Solving (ax-1)² + (bx-2)² = 5x²-8x+5

Question

Fill in the blanks:

(?×x1)2+(?×x2)2=5x28x+5 (?\times x-1)^2+(?\times x-2)^2=5x^2-8x+5

Video Solution

Solution Steps

00:00 Complete the missing
00:03 Mark A and B as unknowns
00:10 Use shortened multiplication formulas to expand the brackets
00:20 Collect like terms
00:34 Factor out common terms from the brackets
00:48 Equate the equal coefficients
01:00 Now we have a system of equations, let's solve for the unknowns
01:03 Let's start with the second equation
01:08 This is the expression for A in terms of B
01:12 Substitute this expression in the first equation to find B
01:21 Use shortened multiplication formulas to expand the brackets
01:31 Collect like terms and arrange the equation
01:49 Use shortened multiplication formulas and find the solution for B
01:52 Substitute this solution to find A
01:57 And this is the solution to the problem

Step-by-Step Solution

To solve this problem, we'll fill in the blanks in the expression (?×x1)2+(?×x2)2=5x28x+5(?\times x-1)^2+(?\times x-2)^2=5x^2-8x+5.

Let's expand the individual expressions:
1. (ax1)2=a2x22ax+1(a \cdot x - 1)^2 = a^2x^2 - 2ax + 1
2. (bx2)2=b2x24bx+4(b \cdot x - 2)^2 = b^2x^2 - 4bx + 4

Now, add these expansions together:
(a2x22ax+1)+(b2x24bx+4)(a^2x^2 - 2ax + 1) + (b^2x^2 - 4bx + 4)
= (a2+b2)x2(2a+4b)x+5(a^2 + b^2)x^2 - (2a + 4b)x + 5

We equate this to the given quadratic: 5x28x+55x^2 - 8x + 5.

Matching coefficients, we have:
a2+b2=5a^2 + b^2 = 5
2a+4b=82a + 4b = 8

Now, solve these equations:

From the second equation: 2a+4b=82a + 4b = 8, divide by 2:
a+2b=4a + 2b = 4

Substitute a=42ba = 4 - 2b from this into a2+b2=5a^2 + b^2 = 5:

(42b)2+b2=5(4 - 2b)^2 + b^2 = 5
1616b+4b2+b2=516 - 16b + 4b^2 + b^2 = 5
5b216b+16=55b^2 - 16b + 16 = 5
5b216b+11=05b^2 - 16b + 11 = 0

Factoring the quadratic equation 5b216b+11=05b^2 - 16b + 11 = 0:
(5b11)(b1)=0(5b - 11)(b - 1) = 0

This gives us b=115b = \frac{11}{5} or b=1b = 1.

Using b=1b = 1, substitute back to find aa:
a=42(1)=2a = 4 - 2(1) = 2

Thus, the correct integers for the blanks are a=2a = 2 and b=1b = 1.

Therefore, the correct answer is 2, 12, \text{ } 1.

Answer

2, 1 2,\text{ }1