Example Question - simplifying algebraic expressions
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Expanding and Simplifying Algebraic Expressions
3p(p - q) - (2p - q)² Step 1: Distribute 3p in the first term = 3p² - 3pq Step 2: Expand the square in the second term = (2p - q)(2p - q) = 4p² - 2pq - 2pq + q² Step 3: Combine like terms in the second term = 4p² - 4pq + q² Step 4: Subtract the expanded second term from the first term = (3p² - 3pq) - (4p² - 4pq + q²) Step 5: Distribute the subtraction across each term in the parentheses = 3p² - 3pq - 4p² + 4pq - q² Step 6: Combine like terms = -p² + pq - q² The final answer: = -p² + pq - q²
Simplifying an Algebraic Expression
The image shows an algebraic expression that needs to be simplified: \[ \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}} + \frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}} \] To simplify this expression, we can combine the two fractions into a single fraction by finding a common denominator, which is \((\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})\). First, we recognize that \((\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})\) equals \(\sqrt{5}^2 - \sqrt{3}^2\), which simplifies to \(5 - 3\), or \(2\). Now, let's work out the combined numerator: For denominator \(\sqrt{5} - \sqrt{3}\), the corresponding numerator is \((1 - \sqrt{2})(\sqrt{5} + \sqrt{3})\). For denominator \(\sqrt{5} + \sqrt{3}\), the corresponding numerator is \((1 + \sqrt{2})(\sqrt{5} - \sqrt{3})\). Let's find each product for the numerators separately: - \((1 - \sqrt{2})(\sqrt{5} + \sqrt{3}) = \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}\) - \((1 + \sqrt{2})(\sqrt{5} - \sqrt{3}) = \sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6}\) Next, combine these to get the new numerator: \[ (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}) + (\sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6}) \] Simplify by cancelling out the opposite terms \( \sqrt{3} \) and \( -\sqrt{3} \), and by combining like terms: \[ 2\sqrt{5} - 2\sqrt{6} \] With the denominator found to be \(2\), we now have: \[ \frac{2\sqrt{5} - 2\sqrt{6}}{2} \] Divide both terms in the numerator by the denominator to simplify further: \[ \sqrt{5} - \sqrt{6} \] Thus, the simplified expression is: \[ \sqrt{5} - \sqrt{6} \]
Solving Algebraic Expression with Squares and Roots
Baiklah, kita akan menyelesaikan soal yang terdapat pada gambar. Ekspresi yang diberikan adalah: \[ 13 \cdot AB + 2 \cdot BC - ( \sqrt{BA} - BC)^2 \] Untuk menyelesaikan ekspresi ini, pertama-tama kita akan ekspansi kuadrat pada bagian \(( \sqrt{BA} - BC)^2 \). Ini akan mengikuti identitas aljabar \( (a - b)^2 = a^2 - 2ab + b^2 \), dengan \(a = \sqrt{BA}\) dan \(b = BC\): \[ ( \sqrt{BA} - BC)^2 = (\sqrt{BA})^2 - 2(\sqrt{BA})(BC) + (BC)^2 \] \[ = BA - 2BC\sqrt{BA} + BC^2 \] Sekarang kita akan substitusi nilai ini ke dalam ekspresi awal: \[ 13 \cdot AB + 2 \cdot BC - (BA - 2BC\sqrt{BA} + BC^2) \] \[ = 13 \cdot AB + 2 \cdot BC - BA + 2BC\sqrt{BA} - BC^2 \] Kita harus berhati-hati dengan tanda negatif yang mengawali kurung. Tanda negatif ini akan mengubah tanda semua suku di dalamnya setelah kurung dihilangkan. Kemudian, kita dapat menyederhanakan ekspresi dengan menggabungkan suku-suku sejenis: 1. Suku \(AB\) atau \(BA\) (karena dalam matematika, \(AB = BA\)) \[ 13AB - BA = 12AB \] 2. Suku \(BC\) \[ 2BC + 2BC\sqrt{BA} - BC^2 \] Menyatukan kedua hasil ini, ekspresi akhirnya adalah: \[ 12AB + 2BC + 2BC\sqrt{BA} - BC^2 \] Ini adalah versi yang disederhanakan dari ekspresi yang diberikan. Catatan penting: Karena tidak ada nilai konkret untuk \(AB\) dan \(BC\), kita tidak dapat menyederhanakan lebih jauh atau mencari nilai numerik tanpa informasi tambahan.
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