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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Here are the solutions to your assignment questions: 1a. Solve 5x^2 - 7x - 6 = 0. Step 1: Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a with a=5, b=-7, c=-6. x = -(-7) ± sqrt((-7)^2 - 4(5)(-6))2(5) Step 2: Simplify the expression under the square root. x = 7 ± sqrt(49 + 120)10 x = 7 ± sqrt(169)10 Step 3: Calculate the square root and find the two solutions. x = (7 ± 13)/(10) x_1 = (7 + 13)/(10) = (20)/(10) = 2 x_2 = (7 - 13)/(10) = (-6)/(10) = -(3)/(5) The solutions are x = 2, -(3)/(5). 1b. Solve 4x^2 + 2x + 3 = 0. Step 1: Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a with a=4, b=2, c=3. x = -2 ± sqrt(2^2 - 4(4)(3))2(4) Step 2: Simplify the expression under the square root. x = -2 ± sqrt(4 - 48)8 x = -2 ± sqrt(-44)8 Step 3: Simplify the square root involving the imaginary unit i. x = -2 ± sqrt(4 × -11)8 x = -2 ± 2isqrt(11)8 Step 4: Simplify the fraction. x = 2(-1 ± isqrt(11))8 x = -1 ± isqrt(11)4 The solutions are x = -1 ± isqrt(11)4. 2a. Convert 101101_2 to base ten. Step 1: Write out the place values for each digit in base 2. 101101_2 = 1 × 2^5 + 0 × 2^4 + 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 1 × 2^0 Step 2: Calculate the value of each term. = 1 × 32 + 0 × 16 + 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 32 + 0 + 8 + 4 + 0 + 1 Step 3: Sum the values. = 45 The base ten equivalent is 45_10. 2b. Convert 156_10 to base two. Step 1: Divide 156 by 2 and record the remainder. Continue dividing the quotient by 2 until the quotient is 0. 156 ÷ 2 = 78 R 0 78 ÷ 2 = 39 R 0 39 ÷ 2 = 19 R 1 19 ÷ 2 = 9 R 1 9 ÷ 2 = 4 R 1 4 ÷ 2 = 2 R 0 2 ÷ 2 = 1 R 0 1 ÷ 2 = 0 R 1 Step 2: Read the remainders from bottom to top. The base two equivalent is 10011100_2. 3a. Evaluate sqrt(50) + sqrt(18). Step 1: Simplify each square root by finding perfect square factors. sqrt(50) = sqrt(25 × 2) = sqrt(25) × sqrt(2) = 5sqrt(2) sqrt(18) = sqrt(9 × 2) = sqrt(9) × sqrt(2) = 3sqrt(2) Step 2: Substitute the simplified terms back into the expression. 5sqrt(2) + 3sqrt(2) Step 3: Combine the like terms. (5+3)sqrt(2) = 8sqrt(2) The evaluated expression is 8sqrt(2). 3b. Simplify sqrt(12) - 2sqrt(27). Step 1: Simplify each square root. sqrt(12) = sqrt(4 × 3) = sqrt(4) × sqrt(3) = 2sqrt(3) sqrt(27) = sqrt(9 × 3) = sqrt(9) × sqrt(3) = 3sqrt(3) Step 2: Substitute the simplified terms back into the expression. 2sqrt(3) - 2(3sqrt(3)) Step 3: Perform the multiplication and combine the like terms. 2sqrt(3) - 6sqrt(3) = (2-6)sqrt(3) = -4sqrt(3) The simplified expression is -4sqrt(3). 4a. Simplify (3a)^3 × (9a^5)^-1. Step 1: Apply the exponent rule (xy)^n = x^n y^n to the first term. (3a)^3 = 3^3 a^3 = 27a^3 Step 2: Apply the exponent rule x^-n = (1)/(x^n) to the second term. (9a^5)^-1 = (1)/(9a^5) Step 3: Multiply the simplified terms. 27a^3 × (1)/(9a^5) = (27a^3)/(9a^5) Step 4: Simplify the fraction using exponent rule (x^m)/(x^n) = x^m-n. = (27)/(9) × a^3-5 = 3a^-2 = (3)/(a^2) The simplified expression is (3)/(a^2). 4b. Simplify 2 - (3)/(x+2) + (4)/((x+3)^2). Step 1: Find a common denominator, which is (x+2)(x+3)^2. 2 - (3)/(x+2) + (4)/((x+3)^2) = (2(x+2)(x+3)^2)/((x+2)(x+3)^2) - (3(x+3)^2)/((x+2)(x+3)^2) + (4(x+2))/((x+2)(x+3)^2) Step 2: Combine the numerators over the common denominator. = (2(x+2)(x+3)^2 - 3(x+3)^2 + 4(x+2))/((x+2)(x+3)^2) Step 3: Expand (x+3)^2 = x^2 + 6x + 9 and then expand the numerator. = (2(x+2)(x^2+6x+9) - 3(x^2+6x+9) + 4x+8)/((x+2)(x+3)^2) = (2(x^3+6x^2+9x+2x^2+12x+18) - 3x^2-18x-27 + 4x+8)/((x+2)(x+3)^2) = (2(x^3+8x^2+21x+18) - 3x^2-18x-27 + 4x+8)/((x+2)(x+3)^2) = (2x^3+16x^2+42x+36 - 3x^2-18x-27 + 4x+8)/((x+2)(x+3)^2) Step 4: Combine like terms in the numerator. = (2x^3 + (16-3)x^2 + (42-18+4)x + (36-27+8))/((x+2)(x+3)^2) = (2x^3 + 13x^2 + 28x + 17)/((x+2)(x+3)^2) The simplified expression is (2x^3 + 13x^2 + 28x + 17)/((x+2)(x+3)^2). 5a. Simplify (2x)/(x+y) + (3y)/(x+y). Step 1: Since the denominators are the same, add the numerators. = (2x+3y)/(x+y) The simplified expression is (2x+3y)/(x+y). 5b. If 2x+5=17 find the value of x. Step 1: Subtract 5 from both sides of the equation. 2x + 5 - 5 = 17 - 5 2x = 12 Step 2: Divide both sides by 2. (2x)/(2) = (12)/(2) x = 6 The value of x is 6. That's 2 down. 3 left today — send the next one.