Solution: We are given that $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $. We are to find $ |a^4 + b^4| $.
["Title: Solving for $ |a^4 + b^4| $ Given $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $", "When presented with the equations $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $, many wonder: What is $ |a^4 + b^4| $? This mathematical challenge combines fundamental trigonometric identità and algebraic manipulation, making it a compelling problem for students of algebra, trigonometry, and complex numbers.", "In this article, we will systematically solve for $ |a^4 + b^4| $ using known relationships and identities.", "---", "### Step 1: Use a Known Identity", "We recall the identity:", "$$\na^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2\n$$", "We are given $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $, so $ a^2b^2 = (ab)^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4} $.", "Now substitute into the identity:", "$$\na^4 + b^4 = (1)^2 - 2 \cdot \frac{1}{4} = 1 - \frac{1}{2} = \frac{1}{2}\n$$", "Thus,", "$$\n|a^4 + b^4| = \left| \frac{1}{2} \right| = \frac{1}{2}\n$$", "---", "### Step 2: Confirm Validity Using Algebraic Relations", "To deepen understanding, let’s confirm this result using substitution based on the given constraints.", "From $ a^2 + b^2 = 1 $, and $ ab = \frac{1}{2} $, consider $ a $ and $ b $ as roots of a quadratic equation:", "$$\nx^2 - sx + p = 0\n$$", "where $ s = a + b $, $ p = ab = \frac{1}{2} $.", "We also know:", "$$\n(a + b)^2 = a^2 + b^2 + 2ab = 1 + 2 \cdot \frac{1}{2} = 2 \Rightarrow s^2 = 2 \Rightarrow s = \pm\sqrt{2}\n$$", "Now, $ a $ and $ b $ are roots of $ x^2 - \sqrt{2}x + \frac{1}{2} = 0 $ or $ x^2 + \sqrt{2}x + \frac{1}{2} = 0 $, depending on the sign. Either way, $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $ are consistent.", "Working with $ a^2 + b^2 = 1 $, we proceed with the earlier identity — which holds regardless of specific values of $ a $ and $ b $ — to compute $ a^4 + b^4 $. Hence, the result $ \frac{1}{2} $ is confirmed valid.", "---", "### Step 3: Why This Matters", "Understanding expressions like $ a^4 + b^4 $ is essential in trigonometry, complex numbers, and optimization problems. When $ a $ and $ b $ represent squares of real or complex numbers (as in $ a^2, b^2 $), this identity helps simplify higher powers efficiently — saving time and reducing complexity.", "---", "### Final Answer", "$$\n\boxed{|a^4 + b^4| = \frac{1}{2}}\n$$", "---", "Keywords: $ a^2 + b^2 = 1 $, $ ab = \frac{1}{2} $, $ a^4 + b^4 $, algebraic identities, trigonometric identities, mathematical problem solving, absolute value, complex numbers, quadratic roots.", "Meta Description:\nGiven $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $, compute $ |a^4 + b^4| $ using algebraic identities. Step-by-step solution with verification and real-world relevance. #Math #Algebra #Trigonometry #MathConfirmation"]
