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Factoring quadratics is a way to express a quadratic equation as a product of two or more simpler expressions. It is an important skill because it allows us to find the solutions, or roots, of the quadratic equation easily. For example, in the equation \(4x^2 - 5x = 0\), we can factor by finding common factors in the terms.

Here, both terms \(4x^2\) and \(-5x\) contain \(x\) as a factor. By factoring \(x\) out of the expression, we simplify the equation to \(x(4x - 5) = 0\).

To successfully factor a quadratic:

• Identify the common factor between the terms. In this example, it's \(x\).
• Divide each term by the common factor and write down the remaining expression within parentheses.
• Verify by expanding to check if the factored form multiplies back to the original quadratic equation.

Setting equations to zero is a crucial initial step in solving quadratic equations. When a quadratic equation equals zero, it becomes easier to apply techniques such as factoring.

In our example, \(4x^2 = 5x\), we start by subtracting \(5x\) from both sides to obtain \(4x^2 - 5x = 0\). By setting the equation to zero, we align ourselves to find solutions using methods like factoring or the quadratic formula later. The logic behind it lies in the zero product property, which says if a product of two terms equals zero, then at least one of the terms must be zero.

This step is fundamental for:

• Simplifying expressions to allow usage of specific techniques like factoring or graphing.
• Applying the zero product property directly to find solutions.

Finding solutions of equations involves identifying the values of \(x\) that satisfy the equation. For a factored equation like \(x(4x - 5) = 0\), the task is straightforward because we use the zero product property.

This property states that if a multiplication of terms results in zero, then any one of those terms could be zero. For our factored equation, this means we set each factor to zero separately:

• \(x = 0\)
• \(4x - 5 = 0\), which simplifies to \(x = \frac{5}{4}\) by adding 5 to both sides and dividing by 4.

Both solutions, \(x = 0\) and \(x = \frac{5}{4}\), make the original equation true.

Remember: Finding solutions is about addressing each possible set equation and solving for \(x\). Consider checking your solutions by substituting them back into the original equation to verify correctness.