Solve for \( x \): \( 2x^2 - 16x + 30 = 0 \).

["How to Solve the Quadratic Equation ( 2x^2 - 16x + 30 = 0 ): A Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, data analysts, and anyone working with mathematical modeling. One common equation students encounter is:", "[\n2x^2 - 16x + 30 = 0\n]", "This article breaks down how to solve this equation step-by-step using the standard quadratic formula, ensuring clarity and ease of understanding. We’ll also explore tips to simplify solving, interpret the results, and apply the solution in real-world scenarios.", "---", "### Step 1: Simplify the Equation", "Before applying the quadratic formula, simplify the equation as much as possible. Notice all terms are divisible by 2:", "[\n\frac{2x^2 - 16x + 30}{2} = 0 \quad \Rightarrow \quad x^2 - 8x + 15 = 0\n]", "Now, we solve the simplified equation:", "[\nx^2 - 8x + 15 = 0\n]", "---", "### Step 2: Identify Coefficients for the Quadratic Formula", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "From ( x^2 - 8x + 15 = 0 ), we identify:", "- ( a = 1 )\n- ( b = -8 )\n- ( c = 15 )", "---", "### Step 3: Apply the Quadratic Formula", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute ( a = 1 ), ( b = -8 ), and ( c = 15 ):", "[\nx = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(15)}}{2(1)}\n]", "Simplify step-by-step:", "[\nx = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm \sqrt{4}}{2}\n]", "[\nx = \frac{8 \pm 2}{2}\n]", "---", "### Step 4: Calculate the Two Solutions", "Now compute both values:", "1. ( x = \frac{8 + 2}{2} = \frac{10}{2} = 5 )", "2. ( x = \frac{8 - 2}{2} = \frac{6}{2} = 3 )", "---", "### Step 5: Write the Final Answer", "The solutions to the equation ( 2x^2 - 16x + 30 = 0 ) are:", "[\n\boxed{x = 3} \quad \ ext{and} \quad \boxed{x = 5}\n]", "---", "### Why These Solutions Matter", "Solving quadratics like this appears in physics (motion models), engineering (structural analysis), economics (profit calculations), and more. The roots ( x = 3 ) and ( x = 5 ) may represent critical points, break-even values, or equilibrium states depending on the context.", "---", "### Quick Recap: Quick Solution Using Factoring", "Because our simplified equation ( x^2 - 8x + 15 = 0 ) has integer coefficients and factors neatly, you can factor it directly:", "Look for two numbers that multiply to 15 and add to -8:\nThese are -3 and -5.", "[\nx^2 - 8x + 15 = (x - 3)(x - 5) = 0\n]", "Set each factor to zero:", "- ( x - 3 = 0 \Rightarrow x = 3 )\n- ( x - 5 = 0 \Rightarrow x = 5 )", "Same results, faster than the quadratic formula when factoring is easy!", "---", "### Summary", "- Always simplify equations when possible.\n- Use the quadratic formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).\n- Identify ( a ), ( b ), and ( c ) correctly.\n- Calculate discriminant (( b^2 - 4ac )) to check solution type:\n - Positive: two real solutions\n - Zero: one real solution\n - Negative: two complex solutions\n- Factoring speeds up solving for simple quadratics.\n- Apply solutions in context—whether modeling or real-world problems.", "Mastering these steps ensures confidence in solving quadratic equations and builds a strong foundation for advanced math topics.", "---", "Keywords: solve (2x^2 - 16x + 30 = 0), quadratic equation solution, quadratic formula, algebraic methods, vertex form, factoring, real-world applications."]