Solving Quadratic Equations with Algebraic Manipulation Techniques

Quadratic equations, often seen in algebraic problems, can be solved using various methods, such as algebraic manipulation. In this discussion, we will explore a specific problem and solve it step by step using various algebraic techniques. This method will help us verify the solutions and understand the underlying mathematical concepts.

Problem Statement

The given equation is:

[ x-1^2 x-3^25 ]

Solution 1: Expand and Simplify

Let's start by expanding the brackets and simplifying the equation:

Step 1: Expand the brackets using the "perfect squares" expansion method

The perfect squares expansion method states that:

[ (a-b^2) a^2 - 2ab b^2 ]

Applying this method:

[ x^2 - 2x 1 (x-1)^2 ]

and

[ x^2 - 2x cdot 3 9 (x-3)^2 ]

So, the given equation [ (x-1)^2 (x-3)^2 5 ] can be rewritten as:

[ x^2 - 2x 1 x^2 - 6x 9 5 ]

Step 2: Simplify the equation

Moving all terms to one side:

[ x^2 - 2x 1 - (x^2 - 6x 9) - 5 0 ]

[ x^2 - 2x 1 - x^2 6x - 9 - 5 0 ]

[ 4x - 13 0 ]

[ 4x 13 ]

[ x frac{13}{4} ]

Solution 2: Subtract and Factorize

Alternatively, we can subtract and factorize the equation:

Step 1: Expand the brackets directly

[ (x-1)^2 (x-3)^2 5 ]

Step 2: Use the difference of squares formula

[ x^2 - 2x 1 - (x^2 - 6x 9) - 5 0 ]

[ x^2 - 2x 1 - x^2 6x - 9 - 5 0 ]

[ 4x - 13 0 ]

Step 3: Solve for x

[ 4x 13 ]

[ x frac{13}{4} ]

Solution 3: Arithmetic Substitution

We can also use the substitution method:

Step 1: Let ( a x - 2 )

[ (a 1)^2 - (a-1)^2 5 ]

Step 2: Use the difference of squares formula

[ (a 1 a-1) (a 1 - (a-1)) 5 ]

[ 2a cdot 2 5 ]

[ 4a 5 ]

[ a frac{5}{4} ]

[ x 2 frac{5}{4} frac{13}{4} ]

Verification

To verify the solution, let's substitute ( x frac{13}{4} ) back into the original equation:

Left side:

[ left( frac{13}{4} - 1 right)^2 left( frac{9}{4} right)^2 frac{81}{16} ]

Right side:

[ left( frac{13}{4} - 3 right)^2 5 left( frac{-1}{4} right)^2 5 frac{1}{16} 5 frac{81}{16} ]

Since both sides are equal, the solution is verified.

Conclusion

Through various algebraic manipulation techniques, we have successfully solved the given quadratic equation. Understanding these methods not only helps in solving problems but also in grasping the underlying algebraic principles.