91Ó°ÊÓ

Solving equations is a fundamental skill in mathematics. It involves finding the value(s) of variable(s) that make an equation true. In the problem above, solving was done by simplifying and factoring. Initially, the equation \(2x(x - 3) = 5x^2 - 7x\) was simplified by distributing and moving all terms to one side, forming the equation \(-3x^2 + 13x = 0\). This step ensured that the equation was set to zero, which is essential for factoring.

Why set the equation to zero? Because equations of the form \(ax^2 + bx + c = 0\) are easier to solve by factoring. Once the equation was set to zero, it was rewritten in a form that could be factored easily. Isolating terms to one side prepares the equation for the factoring process, allowing you to find the values of \(x\) that satisfy the equation.

Polynomials are algebraic expressions that include one or more terms. Each term is composed of variables, coefficients, and exponents. In this exercise, the equation involved polynomial terms. The original equation \(2x(x - 3) = 5x^2 - 7x\) contained quadratic terms, \(2x^2\) and \(5x^2\).

Understanding polynomials helps in recognizing patterns for factoring and simplifying expressions. In our example, after simplifying and rearranging, we processed the polynomial \(-3x^2 + 13x = 0\). Recognizing this as a quadratic form leads to identifying common factors, which is an essential skill in manipulating polynomial expressions. Quadratics often can be simplified to a product of linear factors, which makes solving easier.

Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks for more complex equations and are central to solving and understanding polynomials.

In the exercise, you worked with different expressions on each side of the initial equation. The expression \(2x(x - 3)\) on the left required expanding through distribution. This ultimately facilitated moving all terms to one side and identifying common factors.

Factoring is a way to break down expressions into simpler multiplicative components. For \(-3x^2 + 13x\), we extracted \(x\) as a common factor, leading to the expression \(-3x(x - \frac{13}{3}) = 0\). By breaking down the expression in this way, it clearly revealed the solutions for \(x\). Understanding how to manipulate algebraic expressions is key to unraveling even the most complex equations.