91Ó°ÊÓ

Factoring is a powerful method to solve quadratic equations, but first, you must identify if it's applicable. The key is to express the quadratic equation in a product of simpler expressions set to zero. Let's take the quadratic \[3x^2 - 15x + 18 = 0\]as an example. Before factoring, make sure your equation is in standard form \(ax^2 + bx + c = 0\). This ensures you're ready to look for a common factor or factor pairs. * **Step 1: Identify the Greatest Common Factor (GCF).** - The GCF among terms \(3x^2\), \(-15x\), and \(18\) is \(3\). Factoring this out simplifies the equation to \[3(x^2 - 5x + 6) = 0\]. * **Step 2: Factor Quadratic Expression Fully.** - The new expression inside the parenthesis \(x^2 - 5x + 6\) needs further factorization. Search for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). - These numbers are \(-2\) and \(-3\). Therefore, the expression factors to \[(x - 2)(x - 3) = 0\]. Factoring is often straightforward and offers insight into the characteristics of the quadratic equation, helping find the roots efficiently.

When factoring isn't feasible or if you prefer a more general approach, the quadratic formula serves as a universal tool to find the roots of any quadratic equation. The formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], and it's applicable when the equation is in the standard form \(ax^2 + bx + c = 0\). In our case: - Convert the equation if necessary, but from \(3x^2 - 15x + 18 = 0\), the coefficients are already clear: \(a = 3\), \(b = -15\), \(c = 18\). To apply the quadratic formula: - Calculate the discriminant: \(b^2 - 4ac = (-15)^2 - 4 \cdot 3 \cdot 18 = 225 - 216 = 9\).- Since the discriminant is positive (9), there are two real roots.- Substitute into the quadratic formula: \[x = \frac{15 \pm \sqrt{9}}{6}\]. By calculation, this results in:- \(x = \frac{15 + 3}{6} = 3\)- \(x = \frac{15 - 3}{6} = 2\)The quadratic formula provides a reliable method for solving any quadratic equation without first needing to factor.

The roots of a quadratic equation, also known as solutions or zeros, are the values of \(x\) that make the equation true. They represent where the parabola described by the quadratic equation intersects the x-axis. Substituting each root into the original equation is a helpful way to verify that they indeed solve the equation. In our original example:- **Root Verification for \(x = 2\):** - Substitute \(x = 2\) into the original equation: \[3(2)^2 + 18 = 15 \times 2\]. - Simplifies to \[12 + 18 = 30\], confirming \(x = 2\) is a root.- **Root Verification for \(x = 3\):** - Substitute \(x = 3\) into the original equation: \[3(3)^2 + 18 = 15 \times 3\]. - Simplifies to \[27 + 18 = 45\], confirming \(x = 3\) is a root.Understanding the roots is crucial because they provide insight into the graphical representation of the equation and can work as checkpoints to ensure accuracy in your calculations.