This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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The problem asks to solve the equation: 2^x × 13^x+10 (2^x^2) × 36^x^2 We will assume the symbol represents an equals sign (=), as is common in mathematical problems of this type. Step 1: Rewrite the equation and simplify the right side. The equation is: 2^x × 13^x+10 = (2^x^2) × 36^x^2 First, express 36 in terms of its prime factors: 36 = 6^2 = (2 × 3)^2 = 2^2 × 3^2. Now substitute this into the right side of the equation: 2^x × 13^x+10 = 2^x^2 × (2^2 × 3^2)^x^2 Apply the exponent rule (a^m)^n = a^mn and (ab)^n = a^n b^n: 2^x × 13^x+10 = 2^x^2 × 2^2x^2 × 3^2x^2 Combine the terms with base 2 on the right side using the rule a^m × a^n = a^m+n: 2^x × 13^x+10 = 2^x^2 + 2x^2 × 3^2x^2 2^x × 13^x+10 = 2^3x^2 × 3^2x^2 Step 2: Take the logarithm of both sides. To solve for x, we take the logarithm (e.g., base 10 logarithm, denoted as ) of both sides. Use the logarithm properties (ab) = a + b and (a^b) = b a: (2^x × 13^x+10) = (2^3x^2 × 3^2x^2) x 2 + (x+10) 13 = 3x^2 2 + 2x^2 3 Step 3: Rearrange the equation into a standard quadratic form Ax^2 + Bx + C = 0. Expand the terms: x 2 + x 13 + 10 13 = 3x^2 2 + 2x^2 3 Move all terms to one side to form a quadratic equation: (3 2 + 2 3)x^2 - ( 2 + 13)x - 10 13 = 0 Use logarithm properties a + b = (ab) and b a = (a^b): ((2^3) + (3^2))x^2 - ((2 × 13))x - 10 13 = 0 ( 8 + 9)x^2 - ( 26)x - 10 13 = 0 ((8 × 9))x^2 - ( 26)x - 10 13 = 0 ( 72)x^2 - ( 26)x - 10 13 = 0 This is a quadratic equation where A = 72, B = - 26, and C = -10 13. Step 4: Apply the quadratic formula to find x. The quadratic formula is x = -B ± sqrt(B^2 - 4AC)2A. x = -(- 26) ± sqrt((- 26)^2 - 4( 72)(-10 13))2( 72) x = 26 ± sqrt(( 26)^2 + 40 ( 72)( 13))2 72 This is the exact solution for x. Step 5: Calculate the numerical values (using base-10 logarithms, rounded to 4 decimal places). 72 ≈ 1.8573 26 ≈ 1.4150 13 ≈ 1.1139 Substitute these values into the quadratic formula: A ≈ 1.8573 B ≈ -1.4150 C ≈ -10 × 1.1139 = -11.1390 Calculate the discriminant = B^2 - 4AC: ≈ (-1.4150)^2 - 4(1.8573)(-11.1390) ≈ 2.0022 + 82.7285 ≈ 84.7307 sqrt() ≈ sqrt(84.7307) ≈ 9.2050 Now, calculate the two possible values for x: x = (1.4150 ± 9.2050)/(2(1.8573)) x = (1.4150 ± 9.2050)/(3.7146) x_1 = (1.4150 + 9.2050)/(3.7146) = (10.6200)/(3.7146) ≈ 2.8590 x_2 = (1.4150 - 9.2050)/(3.7146) = (-7.7900)/(3.7146) ≈ -2.0972 The solutions for x are approximately: x ≈ 2.8590 or x ≈ -2.0972 Send me the next one 📸