91Ó°ÊÓ

The distributive property is a fundamental concept in mathematics. It allows us to multiply a single term by a sum or difference of terms inside a parenthesis. Instead of calculating everything inside first, it's often more efficient to 'distribute' the multiplication over each term separately. In this way, each term inside the brackets is multiplied by the term outside the bracket.

In our example, we see the expression \(2k(k-1)\). To apply the distributive property here, we distribute \(2k\) across both parts inside the parenthesis. This results in:

• \(2k \times k = 2k^2\)
• \(2k \times (-1) = -2k\)

Thus, the original expression \(2k(k-1)\) expands to \(2k^2 - 2k\).
This crucial step helps in simplifying and breaking down complex algebraic expressions into more manageable pieces.

Once we've expanded the expression using the distributive property, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same powers. In algebra, combining these like terms simplifies the expression, making it easier to work with further.

In the given problem, we started with the equation: \(6 + 2k^2 - 2k = 5k\). To combine like terms, notice that we have terms with \(k\), specifically \(-2k\) and \(-5k\). These can be combined together because they both contain the same variable raised to the same power:

• Add \(-2k\) with \(-5k\) to get \(-7k\)

Therefore, after this step, the equation transforms into \(2k^2 - 7k + 6 = 0\).
By moving all terms to one side and simplifying, we consolidate the equation into a form that’s easy to analyze or solve.

The standard form of a quadratic equation is a way of writing a quadratic equation in the form \(ax^2 + bx + c = 0\). This form is beneficial because it clearly lays out the terms of the equation and highlights its degree and coefficient relationships.

In the transformed equation \(2k^2 - 7k + 6 = 0\), the terms are already organized according to the standard quadratic form:

• \(a = 2\) (coefficient of \(k^2\))
• \(b = -7\) (coefficient of \(k\))
• \(c = 6\) (the constant term)

By representing quadratic equations in this way, we can easily apply various mathematical methods to find the roots or verify its nature.

This equation, being in standard form, confirms its identity as a quadratic equation since its highest degree term is squared (\(k^2\)). Consequently, we can apply techniques like factoring, completing the square, or using the quadratic formula to find the values of \(k\) that satisfy the equation.