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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Step 1: Simplify the expression 1.1.1. Distribute the -2 to each term inside the parentheses. -2(x-7) = -2(x) - 2(-7) = -2x + 14 The simplified expression is -2x+14. Step 2: Simplify the expression 1.1.2. Expand the binomial squared using the formula (a-b)^2 = a^2 - 2ab + b^2. Here, a=3x and b=5. (3x-5)^2 = (3x)^2 - 2(3x)(5) + (5)^2 = 9x^2 - 30x + 25 The simplified expression is 9x^2-30x+25. Step 3: Simplify the expression 1.1.3. Recognize this as the difference of cubes formula: (a-b)(a^2+ab+b^2) = a^3 - b^3. Here, a=x and b=3. (x-3)(x^2+3x+9) = x^3 - 3^3 = x^3 - 27 The simplified expression is x^3-27. Step 4: Simplify the expression 1.1.4. Recognize this as the difference of squares formula: (a+b)(a-b) = a^2 - b^2. Here, a=(p)/(5) and b=1. ((p)/(5)+1)((p)/(5)-1) = ((p)/(5))^2 - (1)^2 = (p^2)/(25) - 1 The simplified expression is (p^2)/(25)-1. Step 5: Factorise fully 1.2.1. Find the greatest common factor (GCF) of 3p and 15, which is 3. 3p-15 = 3(p) - 3(5) = 3(p-5) The factored expression is 3(p-5). Step 6: Factorise fully 1.2.2. Factor the quadratic trinomial 2x^2-3x-5. We look for two numbers that multiply to 2 × -5 = -10 and add to -3. These numbers are -5 and 2. Rewrite the middle term using these numbers: 2x^2 - 5x + 2x - 5 Group the terms and factor: (2x^2 - 5x) + (2x - 5) x(2x - 5) + 1(2x - 5) (x+1)(2x-5) The factored expression is (x+1)(2x-5). Step 7: Factorise fully 1.2.3. Group the first three terms, which form a perfect square trinomial, and then apply the difference of squares formula. a^2-2ab+b^2-64c^2 = (a^2-2ab+b^2) - 64c^2 = (a-b)^2 - (8c)^2 Now apply the difference of squares formula, X^2 - Y^2 = (X-Y)(X+Y), where X=(a-b) and Y=8c. = ((a-b)-8c)((a-b)+8c) = (a-b-8c)(a-b+8c) The factored expression is (a-b-8c)(a-b+8c). Step 8: Determine the value of m for 1.3.1. Given m = sqrt(b^2 - 4ac) and a=2, b=-3, c=-1. Substitute the values into the formula: m = sqrt((-3)^2 - 4(2)(-1)) m = sqrt(9 - (-8)) m = sqrt(9 + 8) m = sqrt(17) The value of m in simplest surd form is sqrt(17). Step 9: State whether m is rational, irrational or non-real for 1.3.2. Since m = sqrt(17) and 17 is not a perfect square, sqrt(17) cannot be expressed as a simple fraction of two integers. Therefore, m is irrational. The value of m is irrational. Step 10: Determine between which TWO consecutive integers m lies for 1.3.3. We know m = sqrt(17). We need to find two consecutive integers whose squares are just below and just above 17. 4^2 = 16 5^2 = 25 Since 16 < 17 < 25, it follows that sqrt(16) < sqrt(17) < sqrt(25). So, 4 < sqrt(17) < 5. Thus, m lies between the integers 4 and 5. Step 11: Solve for x in 2.1.1. 3x-11=0 Add 11 to both sides: 3x = 11 Divide by 3: x = (11)/(3) The solution is x=(11)/(3). Step 12: Solve for x in 2.1.2. 2x^2-x=6 Rearrange the equation to the standard quadratic form ax^2+bx+c=0: 2x^2-x-6=0 Factor the quadratic equation. We look for two numbers that multiply to 2 × -6 = -12 and add to -1. These numbers are -4 and 3. Rewrite the middle term: 2x^2 - 4x + 3x - 6 = 0 Group the terms and factor: (2x^2 - 4x) + (3x - 6) = 0 2x(x - 2) + 3(x - 2) = 0 (2x+3)(x-2) = 0 Set each factor to zero to find the solutions: 2x+3=0 2x=-3 x=-(3)/(2) x-2=0 x=2 The solutions are x=-(3)/(2) or x=2. Step 13: Solve for x in 2.1.3. 5^x = (1)/(125) Express the right side with a base of 5: 125 = 5^3 So, (1)/(125) = (1)/(5^3) = 5^-3. Now the equation is: 5^x = 5^-3 Since the bases are the same, the exponents must be equal: x = -3 The solution is x=-3. Step 14: Solve for x in the inequality 2.2.1. (x)/(2)+1 < 3(x+7) First, distribute the 3 on the right side: (x)/(2)+1 < 3x+21 To eliminate the fraction, multiply the entire inequality by 2: 2((x)/(2)) + 2(1) < 2(3x) + 2(21) x + 2 < 6x + 42 Subtract x from both sides: 2 < 5x + 42 Subtract 42 from both sides: 2 - 42 < 5x -40 < 5x Divide by 5: (-40)/(5) < x -8 < x The solution to the inequality is x > -8. Step 15: Represent the answer to QUESTION 2.2.1 on a number line for 2.2.2. Draw a number line. Mark -8 on it. Since x > -8, use an open circle at -8 (because x is strictly greater than -8, not equal to it) and draw an arrow extending to the right from -8. ` <---------------------------------------------------------------------> -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 o-----------------------------------------------------> ` The number line representation shows an open circle at -8 and a line extending to the right. Send me the next one 📸