5x^2 + \frac{5}{x^2} = 5 \times 14 = 70

["Understanding the Equation: 5x² + \frac{5}{x²} = 70 – A Step-by-Step Breakdown", "Mathematics often presents equations that seem daunting at first glance, but with the right approach, even complex expressions become manageable. One such equation that combines quadratic forms with reciprocals is:", "[\n5x^2 + \frac{5}{x^2} = 70\n]", "At first look, this equation may appear intimidating due to its algebraic nature and implicit reciprocal terms. However, with strategic manipulation and algebraic insight, we can simplify it and even solve for (x). This article explores the step-by-step solution, explores real-world applications, and explains why recognizing patterns in equations like this is crucial in both algebra and advanced mathematics.", "---", "### Step 1: Simplify the Equation", "Start by dividing both sides of the equation by 5 to reduce complexity:", "[\nx^2 + \frac{1}{x^2} = 14\n]", "This simplified form makes the equation easier to work with. Now the problem becomes solving:", "[\nx^2 + \frac{1}{x^2} = 14\n]", "---", "### Step 2: Let ( y = x^2 )", "To eliminate the denominator, let:", "[\ny = x^2 \quad \ ext{(so } y > 0 \ ext{ since } x^2 > 0\ ext{)}\n]", "Substitute into the equation:", "[\ny + \frac{1}{y} = 14\n]", "Now this is a rational equation in (y), which is simpler to solve.", "---", "### Step 3: Multiply Through by (y) to Eliminate the Denominator", "Multiply every term by (y) (valid since ( y <br/>\neq 0 )):", "[\ny^2 + 1 = 14y\n]", "Rearranging gives a standard quadratic:", "[\ny^2 - 14y + 1 = 0\n]", "---", "### Step 4: Solve the Quadratic Equation", "Use the quadratic formula:", "[\ny = \frac{14 \pm \sqrt{(-14)^2 - 4(1)(1)}}{2} = \frac{14 \pm \sqrt{196 - 4}}{2} = \frac{14 \pm \sqrt{192}}{2}\n]", "Simplify (\sqrt{192}):", "[\n\sqrt{192} = \sqrt{64 \ imes 3} = 8\sqrt{3}\n]", "So:", "[\ny = \frac{14 \pm 8\sqrt{3}}{2} = 7 \pm 4\sqrt{3}\n]", "---", "### Step 5: Recall ( y = x^2 ) and Solve for (x)", "Since ( y = x^2 ), we take square roots:", "[\nx = \pm \sqrt{7 \pm 4\sqrt{3}}\n]", "These expressions involve nesting square roots, which can be simplified using conjugate pairs:", "Observe that:", "[\n\sqrt{7 + 4\sqrt{3}} = \sqrt{a} + \sqrt{b} \quad \ ext{for some } a, b\n]", "Try squaring:\n((\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} = 7 + 4\sqrt{3})", "Match terms:\n- ( a + b = 7 )\n- ( 2\sqrt{ab} = 4\sqrt{3} \Rightarrow \sqrt{ab} = 2\sqrt{3} \Rightarrow ab = 12 )", "Solving (a + b = 7), (ab = 12), we get (a = 3, b = 4) (or vice versa).", "Thus:", "[\n\sqrt{7 + 4\sqrt{3}} = \sqrt{4} + \sqrt{3} = 2 + \sqrt{3}\n]", "Similarly,", "[\n\sqrt{7 - 4\sqrt{3}} = \sqrt{4} - \sqrt{3} = 2 - \sqrt{3} \quad \ ext{(positive since } 2 > \sqrt{3}\ ext{)}\n]", "---", "### Final Solution", "Therefore, the real solutions to the original equation are:", "[\nx = \pm(2 + \sqrt{3}) \quad \ ext{and} \quad x = \pm(2 - \sqrt{3})\n]", "---", "### Why This Equation Matters", "Equations like (5x^2 + \frac{5}{x^2} = 70) appear in physics (e.g., wave interference modelled with energy terms), optimization problems, and even in signal processing. Recognizing symmetry and applying algebraic identities—such as (x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2)—and simplifying nested radicals is crucial both for exact solutions and real-world modeling.", "---", "### Summary", "- Divide by 5 to simplify: (x^2 + \frac{1}{x^2} = 14)\n- Substitute (y = x^2) to form (y + \frac{1}{y} = 14)\n- Solve the quadratic and find (y = 7 \pm 4\sqrt{3})\n- Extract (x = \pm(2 \pm \sqrt{3}))", "Understanding such equations strengthens algebraic dexterity and prepares learners for advanced topics where symmetry and transformation play key roles.", "---", "Try solving: (5x^2 + \frac{5}{x^2} = 70) today—and unlock the beauty of symmetry in algebra!"]