Solve for \( x \) in the equation \( 4x^2 - 12x + 9 = 0 \).

["# Solving the Quadratic Equation ( 4x^2 - 12x + 9 = 0 ): Step-by-Step Guide", "Understanding how to solve quadratic equations is essential in algebra, and mastering this skill unlocks solutions to many real-world problems. In this article, we’ll walk through solving the equation:\n[\n4x^2 - 12x + 9 = 0\n]", "Whether you're a high school student, a student teacher, or someone refreshing your math skills, this guide will clarify how to find the values of ( x ) that satisfy the equation—specifically, solving for ( x ).", "---", "## What is a Quadratic Equation?", "A quadratic equation is any equation that can be written in the standard form:\n[\nax^2 + bx + c = 0\n]\nwhere ( a ), ( b ), and ( c ) are constants and ( a <br/>\ne 0 ). The equation ( 4x^2 - 12x + 9 = 0 ) matches this form with:\n- ( a = 4 )\n- ( b = -12 )\n- ( c = 9 )", "---", "## Step 1: Recognize the Structure of the Equation", "We start by examining whether the quadratic can be factored easily. The equation:\n[\n4x^2 - 12x + 9 = 0\n]\nresembles a perfect square trinomial because:\n- The first term ( 4x^2 = (2x)^2 )\n- The last term ( 9 = 3^2 )\n- The middle term ( -12x = -2 \cdot 2x \cdot 3 )", "Indeed:\n[\n4x^2 - 12x + 9 = (2x - 3)^2\n]", "---", "## Step 2: Rewrite as a Square and Set to Zero", "Using the perfect square insight:\n[\n(2x - 3)^2 = 0\n]", "---", "## Step 3: Solve by Taking the Square Root", "Taking the square root of both sides:\n[\n2x - 3 = 0\n]", "Solving for ( x ):\n[\n2x = 3 \quad \Rightarrow \quad x = \frac{3}{2}\n]", "---", "## Step 4: Verify the Solution", "Since the equation factors perfectly, there’s only one unique solution: ( x = \frac{3}{2} ). This is a repeated root or a double root, meaning the parabola touches the x-axis at exactly one point.", "To confirm:\nSubstitute ( x = \frac{3}{2} ) into the original equation:\n[\n4\left(\frac{3}{2}\right)^2 - 12\left(\frac{3}{2}\right) + 9 = 4 \cdot \frac{9}{4} - 18 + 9 = 9 - 18 + 9 = 0\n]\n✅ The solution satisfies the equation.", "---", "## Alternative Method: Quadratic Formula", "For completeness, we verify using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in ( a = 4 ), ( b = -12 ), ( c = 9 ):\n[\nx = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(9)}}{2(4)} = \frac{12 \pm \sqrt{144 - 144}}{8} = \frac{12 \pm 0}{8} = \frac{12}{8} = \frac{3}{2}\n]", "This confirms our earlier result.", "---", "## Final Answer", "The solution to ( 4x^2 - 12x + 9 = 0 ) is:\n[\n\boxed{x = \frac{3}{2}}\n]", "---", "## Why This Equation Matters", "Beyond mechanics, solving ( 4x^2 - 12x + 9 = 0 ) exemplifies key concepts in algebra:\n- Perfect square trinomials: common in factoring and simplification\n- Repeated roots: illustrate multiplicity in quadratic solutions\n- Efficient solving: recognizing patterns saves time in exams and real problems", "Understanding such equations builds a strong foundation for higher math, including conic sections, calculus, and polynomial modeling.", "---", "## Want More Help Solving Quadratics?", "Need step-by-step guidance with other quadratics? Explore resources on completing the square, discriminant interpretation, and graphing solutions. Mastering quadratic equations opens doors to algebra mastery!", "---", "Tagline for SEO:\nLearn how to solve ( 4x^2 - 12x + 9 = 0 ) easily — perfect square formula, exact solution, step-by-step explanation.", "Keywords: solve ( 4x^2 - 12x + 9 = 0 ), quadratic equation solution, algebra factoring, perfect square trinomial, quadratic formula, repeated root, math tutorial, high school algebra."]