y = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4} = \frac{12 \pm \sqrt{144 + 16}}{8} = \frac{12 \pm \sqrt{160}}{8}

["Solving Quadratic Equations: A Step-by-Step Guide Using the Quadratic Formula", "Understanding how to solve quadratic equations is a crucial skill in algebra and plays a key role in many advanced math applications. One of the most powerful tools for solving quadratics is the quadratic formula, which allows students and learners to find exact solutions quickly and accurately. In this article, we’ll explore the process step-by-step by solving the equation:", "[\ny = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4}\n]", "### What Is the Quadratic Formula?", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "The quadratic formula gives the solutions for ( x ) (or ( y ) in substitution contexts) as:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula works for every quadratic equation, regardless of whether the roots are rational, irrational, or complex.", "### Step-by-Step Demonstration", "Let’s begin with your equation rewritten to clearly show coefficients:", "Given:\n[\ny = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4}\n]", "Step 1: Identify coefficients", "From the standard form ( ax^2 + bx + c = 0 ), match the coefficients:", "- ( a = 4 )\n- ( b = -12 )\n- ( c = -1 )", "Step 2: Substitute coefficients into the quadratic formula", "Substituting ( a ), ( b ), and ( c ):", "[\ny = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4}\n]", "Simplify the numerator:", "- ( -(-12) = 12 )\n- ( (-12)^2 = 144 )\n- ( 4 \cdot 4 \cdot (-1) = -16 ), but since it's subtracted in the discriminant, this becomes ( -(-16) = +16 )", "Thus:", "[\ny = \frac{12 \pm \sqrt{144 + 16}}{8}\n]", "Step 3: Simplify the discriminant", "[\n\sqrt{144 + 16} = \sqrt{160}\n]", "Now factor ( \sqrt{160} ) to simplify:", "[\n160 = 16 \ imes 10 \Rightarrow \sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10}\n]", "Step 4: Final simplified expression", "[\ny = \frac{12 \pm 4\sqrt{10}}{8}\n]", "Step 5: Reduce the fraction (optional)", "Both terms in the numerator share a common factor of 4:", "[\ny = \frac{4(3 \pm \sqrt{10})}{8} = \frac{3 \pm \sqrt{10}}{2}\n]", "### Why Is This Important?", "Knowing how to solve quadratic equations opens the door to modeling real-world scenarios, from projectile motion in physics to maximizing profit in economics. The quadratic formula eliminates guesswork and ensures precision, especially when roots are irrational or complex.", "### Summary", "- The quadratic formula handles any quadratic equation ( ax^2 + bx + c ).\n- Careful identification of coefficients ( a ), ( b ), and ( c ) is key.\n- Simplifying the discriminant and reducing the fraction improves clarity.\n- Example: Solving ( y = \frac{-(-12) \pm \sqrt{160}}{8} ) leads to ( y = \frac{12 \pm \sqrt{160}}{8} = \frac{3 \pm \sqrt{10}}{2} ).", "### Practice Tips", "- Always simplify radicals where possible.\n- Factoring coefficients before applying the formula speeds up calculations.\n- Double-check signs—especially when ( b ) is negative or the product ( 4ac ) is negative.", "Mastering the quadratic formula empowers learners to tackle complex equations with confidence. Whether in the classroom, standardized tests, or real-life applications, this formula is an essential tool in every student’s math toolkit.", "---", "Keywords: quadratic formula, solve quadratics, discriminant, solving square roots, algebra 2, quadratic equations, math tutorial, solving equations steps, discriminant simplification, radical simplification, educational math guide", "Meta Description: Learn how to solve quadratic equations using the quadratic formula with step-by-step examples, including simplification and real-world applications. Understand coefficients, discriminants, and step-by-step transformation."]