["Solving the Quadratic Equation: How to Solve ( 3x^2 - 12x + 9 = 0 )", "Quadratic equations are fundamental in algebra and appear frequently in science, engineering, and economics. One common challenge students face is solving quadratic equations like ( 3x^2 - 12x + 9 = 0 ). In this article, we walk through the step-by-step method to solve this equation, explain key concepts, and highlight useful solving techniques for better understanding and future problem-solving.", "---", "### Understanding the Equation
\nThe given equation is:
\n[
\n3x^2 - 12x + 9 = 0
\n]
\nThis is a standard quadratic equation in the form:
\n[
\nax^2 + bx + c = 0
\n]
\nwhere ( a = 3 ), ( b = -12 ), and ( c = 9 ).", "---", "### Step 1: Simplify the Equation
\nTo make solving easier, first check if the equation can be simplified by dividing all terms by the greatest common divisor (GCD) of the coefficients. Here, GCD of 3, -12, and 9 is 3, so divide everything by 3:
\n[
\nx^2 - 4x + 3 = 0
\n]", "---", "### Step 2: Factor the Quadratic
\nNow, attempt to factor the simplified quadratic:
\n[
\nx^2 - 4x + 3 = 0
\n]
\nWe look for two numbers that multiply to ( +3 ) and add to ( -4 ). These numbers are ( -1 ) and ( -3 ). Thus, the equation factors as:
\n[
\n(x - 1)(x - 3) = 0
\n]", "---", "### Step 3: Apply the Zero Product Property
\nSet each factor equal to zero:
\n[
\nx - 1 = 0 \quad \ ext{or} \quad x - 3 = 0
\n]
\nSolving these gives:
\n[
\nx = 1 \quad \ ext{or} \quad x = 3
\n]", "---", "### Step 4: Verify Using the Quadratic Formula (Optional Check)
\nFor confirmation, use the quadratic formula:
\n[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]
\nWith ( a = 1 ), ( b = -4 ), ( c = 3 ):
\n[
\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)} = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm \sqrt{4}}{2} = \frac{4 \pm 2}{2}
\n]
\nSo,
\n[
\nx = \frac{4 + 2}{2} = 3, \quad x = \frac{4 - 2}{2} = 1
\n]
\nThis matches our earlier results.", "---", "### Why Solving Quadratics Matters
\nQuadratic equations model real-world phenomena such as projectile motion, optimization problems, and area calculations. Mastering techniques like factoring, completing the square, and using the quadratic formula equips students with powerful tools for advanced math and science applications.", "---", "### Summary
\nTo solve ( 3x^2 - 12x + 9 = 0 ):
\n1. Simplify by dividing by common factors
\n2. Factor the simplified equation
\n3. Use zero product property to find roots
\n4. Verify with the quadratic formula", "The solutions are ( x = 1 ) and ( x = 3 ).", "---", "Keywords: solve (3x^2 - 12x + 9 = 0), quadratic equation solutions, factor quadratic equations, use quadratic formula, solve (x^2 - 4x + 3 = 0), step-by-step quadratic problems.
\nMeta description: Learn how to solve (3x^2 - 12x + 9 = 0) easily using factoring, verification with the quadratic formula, and step-by-step methods. Ideal for students mastering algebra."]
