A quadratic equation is given by \( 3x^2 - 12x + 9 = 0 \). Find the roots using the quadratic formula.
["# Solving Quadratic Equations: Finding Roots of ( 3x^2 - 12x + 9 = 0 ) Using the Quadratic Formula", "Learning how to solve quadratic equations is a foundational skill in algebra. Quadratic equations—expressions of the form ( ax^2 + bx + c = 0 )—arise frequently in mathematics, physics, engineering, and many real-world applications. In this article, we’ll explore how to find the roots of the quadratic equation\n[\n3x^2 - 12x + 9 = 0\n]\nusing the quadratic formula, with clear explanations and step-by-step guidance.", "## What Is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation where the highest power of the variable ( x ) is 2. The general form is:\n[\nax^2 + bx + c = 0\n]\nIn our example:\n- ( a = 3 )\n- ( b = -12 )\n- ( c = 9 )", "The solutions (or roots) of this equation represent the values of ( x ) that satisfy the equation, often referred to as the zeros or roots of the quadratic.", "## The Quadratic Formula: A Powerful Tool", "The quadratic formula provides a direct method to find the roots of any quadratic equation:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nThis formula works for all real values of ( a ), ( b ), and ( c ), including cases with complex roots when the discriminant (( b^2 - 4ac )) is negative.", "### Step 1: Identify Coefficients\nFirst, confirm the coefficients from our equation:\n- ( a = 3 )\n- ( b = -12 )\n- ( c = 9 )", "### Step 2: Calculate the Discriminant\nBefore applying the formula, compute the discriminant:\n[\n\Delta = b^2 - 4ac = (-12)^2 - 4(3)(9) = 144 - 108 = 36\n]\nSince ( \Delta = 36 > 0 ), we know there are two distinct real roots.", "### Step 3: Apply the Quadratic Formula\nSubstitute ( a ), ( b ), and ( \Delta ) into the formula:\n[\nx = \frac{-(-12) \pm \sqrt{36}}{2 \ imes 3} = \frac{12 \pm 6}{6}\n]", "This gives two possible solutions:\n[\nx_1 = \frac{12 + 6}{6} = \frac{18}{6} = 3\n]\n[\nx_2 = \frac{12 - 6}{6} = \frac{6}{6} = 1\n]", "### Step 4: Final Answer\nThe roots of the quadratic equation ( 3x^2 - 12x + 9 = 0 ) are:\n[\n\boxed{x = 1 \quad \ ext{and} \quad x = 3}\n]", "## Verification", "To confirm the roots, substitute ( x = 1 ) and ( x = 3 ) back into the original equation:", "- For ( x = 1 ):\n[\n3(1)^2 - 12(1) + 9 = 3 - 12 + 9 = 0\n]", "- For ( x = 3 ):\n[\n3(3)^2 - 12(3) + 9 = 27 - 36 + 9 = 0\n]", "Both values satisfy the equation, confirming the correctness of the roots.", "## Summary", "Solving quadratic equations using the quadratic formula is efficient and reliable. For ( 3x^2 - 12x + 9 = 0 ), the roots are ( x = 1 ) and ( x = 3 ), obtained through careful substitution and calculation. This method applies universally to any quadratic equation, making it an essential tool in algebra.", "Whether you're solving problems in school, engineering, or everyday applications, mastering the quadratic formula empowers you to tackle real-world quadratic relationships with confidence."]
