3x^2 - 4 = 0 \implies x^2 = \frac{4}{3} \implies x = \pm \frac{2}{\sqrt{3}} \approx \pm 1.1547

["How to Solve the Quadratic Equation 3x² - 4 = 0 and Understanding Its Solutions", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and math enthusiasts. One common problem is finding the values of ( x ) that satisfy the equation", "[\n3x^2 - 4 = 0\n]", "This seemingly simple equation unlocks key concepts in algebra, including isolating variables, simplifying expressions, and rationalizing denominators. In this article, we’ll explore step-by-step how to solve ( 3x^2 - 4 = 0 ), derive its exact and approximate solutions, and clarify important algebraic manipulations.", "---", "### Step 1: Isolate the Quadratic Term", "We begin by isolating ( x^2 ) to simplify the equation:", "[\n3x^2 - 4 = 0\n]", "Add 4 to both sides:", "[\n3x^2 = 4\n]", "Now divide both sides by 3:", "[\nx^2 = \frac{4}{3}\n]", "This result is crucial—it expresses ( x^2 ) in its simplest fractional form.", "---", "### Step 2: Take the Square Root of Both Sides", "To solve for ( x ), we取 the square root of both sides:", "[\nx = \pm \sqrt{\frac{4}{3}}\n]", "Taking the square root introduces both the positive and negative possibilities, reflected by the ( \pm ) symbol.", "---", "### Step 3: Simplify the Square Root Expression", "We can simplify ( \sqrt{\frac{4}{3}} ) by separating the numerator and denominator:", "[\n\sqrt{\frac{4}{3}} = \frac{\sqrt{4}}{\sqrt{3}} = \frac{2}{\sqrt{3}}\n]", "However, the radical in the denominator is not preferred in standard algebraic form. To rationalize the denominator, multiply numerator and denominator by ( \sqrt{3} ):", "[\n\frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\n]", "Thus, the full set of solutions becomes:", "[\nx = \pm \frac{2\sqrt{3}}{3} \approx \pm 1.1547\n]", "---", "### Step 4: Exact vs. Approximate Solutions", "While ( \frac{2}{\sqrt{3}} ) is an exact symbolic representation, it’s often useful to express answers with rational denominators. Therefore, the more precise and commonly used form is:", "[\nx = \pm \frac{2\sqrt{3}}{3} \approx \pm 1.1547\n]", "Understanding both forms allows flexibility—exact expressions are critical in algebra and higher mathematics, while decimal approximations aid in practical applications.", "---", "### Why This Equation Matters", "Solving ( 3x^2 - 4 = 0 ) supports several important mathematical ideas:\n- Quadratic Formula Foundation: It’s a linear case of quadratics, laying groundwork for solving ( ax^2 + bx + c = 0 ).\n- Square Roots and Rationalization: Demonstrates manipulating radicals and clearing denominators.\n- Symmetry in Solutions: The plus-minus solution reflects the symmetry inherent in real quadratic roots.", "---", "### Final Note on the Approximate Value", "Although ( \frac{2}{\sqrt{3}} \approx 1.1547 ) is handy for quick calculations, the exact form ensures accuracy in subsequent mathematical operations, especially when used in calculus, geometry, or further algebra.", "---", "Summary:\nStart with ( 3x^2 - 4 = 0 ) → isolate ( x^2 = \frac{4}{3} ) → solve for ( x ) via square root → simplify to ( x = \pm \frac{2\sqrt{3}}{3} ) → approximately ( \pm 1.1547 ).\nMastering this process empowers you to solve more complex quadratics with confidence.", "---", "Keywords: quadratic equation, solve 3x² - 4 = 0, square root manipulation, rationalize denominator, ( x = \pm \frac{2\sqrt{3}}{3} ), algebra basics, solve quadratic equations, exact and approximate solutions."]