How to Factorize and Solve the Quadratic Expression x2 - 2x 0
Factoring and solving quadratic expressions is a fundamental skill in algebra, essential for many advanced mathematical concepts. In this article, we will explore the quadratic expression x2 - 2x 0, detailing the steps to factorize and solve for the variable x. We will also discuss the underlying principles and provide examples to illustrate the process.
Understanding the Quadratic Expression
A quadratic expression is a polynomial of degree 2, generally written in the form ax2 bx c 0. In our case, the expression is x2 - 2x 0, where a 1, b -2, and c 0.
Factorizing the Expression
To factorize the quadratic expression x2 - 2x 0, we need to identify a common factor in the terms of the expression. In this case, the common factor is x. By factoring out x, we rewrite the expression as:
x(x - 2) 0This factorization is based on the distributive property of multiplication over addition, which states that a(b c) ab ac.
Solving for x
Now that the expression is factored, we can solve for x by recognizing that if the product of two factors is zero, then at least one of the factors must be zero. This is known as the zero-product property:
If ab 0, then a 0 or b 0.
Applying this property to our factored expression:
x(x - 2) 0
We get two possible solutions:
x 0 x - 2 0 → x 2Therefore, the solutions to the quadratic equation x2 - 2x 0 are x 0 and x 2.
Verification of Solutions
To verify these solutions, we substitute each value back into the original equation:
x 0:
02 - 2(0) 0 - 0 0
x 2:
22 - 2(2) 4 - 4 0
Both values satisfy the equation, confirming that x 0 and x 2 are indeed the correct solutions.
Conclusion
In conclusion, factoring the quadratic expression x2 - 2x 0 by taking out the common factor x leads us to the equation x(x - 2) 0. The solutions to this factored equation, based on the zero-product property, are x 0 and x 2. By verifying these solutions, we can confirm their accuracy.
Understanding this process is crucial for tackling more complex quadratic equations and other algebraic problems.