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Factoring quadratics is a common method used to solve quadratic equations. Let's break it down in simple terms. Imagine you want to find the numbers that, when multiplied together, will produce your quadratic equation. When factoring, you're essentially breaking up an equation into simpler parts, called factors. Take the quadratic equation we have: \[3x^2 - 8x = 0\]The goal is to express this equation in a form where each part can be set to zero. This process helps us identify the roots or solutions to the equation. In our example, the equation can be factored as: \[x(3x - 8) = 0\]Each factor represents a potential solution. By setting each factor equal to zero, you can solve for the variable \(x\).

• Set \(x = 0\)
• Set \(3x - 8 = 0\)

Factoring is a powerful tool. It allows for quick recognition of solutions, especially when the quadratic equation is easily decomposable into simpler terms.

Expanding squares involves multiplying out a squared expression to simplify or rewrite a quadratic equation. Consider the expression \((3x-4)^2\),which is a perfect square trinomial. Expanding this involves using the formula \((a-b)^2 = a^2 - 2ab + b^2\).Here, substitute \(a = 3x\) and \(b = 4\):\[ (3x)^2 - 2(3x)(4) + 4^2 \]When expanded, it gives:\[ 9x^2 - 24x + 16 \]This step not only simplifies the equation but also prepares it for solving by turning a more complex operation into simpler terms. By expanding squares, you make it easier to see the relationships between terms and prepare an equation for further simplification or solving.

Solving quadratic equations involves various methods, depending on the form and structure of the equation at hand. Once a quadratic equation is presented, like our example:\[9x^2 - 24x = 0\],you can choose how to solve it based on its complexity or the techniques you are comfortable with. For this problem, we simplified and factored which led us to set each factor to zero.

• First, \(x = 0\) directly gives one solution.
• Second, setting \(3x - 8 = 0\) and solving gives another solution: \[x = \frac{8}{3}\]

The advantage of solving by factoring is that it splits the polynomial into simpler linear equations. These can then be easily solved.Different types of quadratics might require other methods like the quadratic formula or completing the square. However, in this example, factoring was straightforward as the expression readily decomposed into linear factors.