%5E2%2B2(x-3)%5E2%3D18-10x.jpeg "(x-1)^2+2(x-3)^2=18-10x")
Solving the Equation: (x-1)^2 + 2(x-3)^2 = 18 - 10x
In this article, we will solve the equation (x-1)^2 + 2(x-3)^2 = 18 - 10x using algebraic methods.
Step 1: Expand the Squares
The equation contains two squared terms: (x-1)^2 and (x-3)^2. Let's expand these squares using the formula (a-b)^2 = a^2 - 2ab + b^2.
(x-1)^2 = x^2 - 2x + 1
(x-3)^2 = x^2 - 6x + 9
Now, substitute these expansions into the original equation:
x^2 - 2x + 1 + 2(x^2 - 6x + 9) = 18 - 10x
Step 2: Simplify the Equation
Combine like terms on the left-hand side of the equation:
x^2 - 2x + 1 + 2x^2 - 12x + 18 = 18 - 10x
Combine the x^2 terms:
3x^2 - 14x + 19 = 18 - 10x
Step 3: Rearrange the Equation
Rearrange the equation to isolate the x terms on one side and the constants on the other:
3x^2 - 14x + 10x + 19 - 18 = 0
3x^2 - 4x + 1 = 0
Step 4: Factor the Quadratic
Try to factor the quadratic expression:
3x^2 - 4x + 1 = (3x - 1)(x - 1) = 0
This tells us that either (3x - 1) = 0 or (x - 1) = 0.
Step 5: Solve for x
Solve for x in each of the above equations:
3x - 1 = 0 --> 3x = 1 --> x = 1/3
x - 1 = 0 --> x = 1
Therefore, the solutions to the equation are x = 1/3 and x = 1.
Conclusion
In this article, we successfully solved the equation (x-1)^2 + 2(x-3)^2 = 18 - 10x using algebraic methods. The solutions to the equation are x = 1/3 and x = 1.
