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 the questions: Question 6.1.1 Determine the sum of 3x^2 + 7x + 7 and 2x^2 + 3x + 4 by adding like terms. Step 1: Write the sum of the two expressions. (3x^2 + 7x + 7) + (2x^2 + 3x + 4) Step 2: Group like terms. (3x^2 + 2x^2) + (7x + 3x) + (7 + 4) Step 3: Add the coefficients of like terms. 5x^2 + 10x + 11 The sum is 5x^2 + 10x + 11. Question 6.1.2 Subtract 2x^2 + 3x + 4 from 3x^2 + 7x + 7. Step 1: Write the subtraction expression. (3x^2 + 7x + 7) - (2x^2 + 3x + 4) Step 2: Distribute the negative sign and group like terms. 3x^2 + 7x + 7 - 2x^2 - 3x - 4 (3x^2 - 2x^2) + (7x - 3x) + (7 - 4) Step 3: Combine like terms. x^2 + 4x + 3 The difference is x^2 + 4x + 3. Question 6.2.1 Expand and simplify (x-2)(x+2). Step 1: Apply the difference of squares formula, (a-b)(a+b) = a^2 - b^2. x^2 - 2^2 Step 2: Simplify the expression. x^2 - 4 The expanded and simplified expression is x^2 - 4. Question 6.2.2 Expand and simplify (x+3)^2. Step 1: Apply the perfect square formula, (a+b)^2 = a^2 + 2ab + b^2. x^2 + 2(x)(3) + 3^2 Step 2: Simplify the terms. x^2 + 6x + 9 The expanded and simplified expression is x^2 + 6x + 9. Question 6.3.1 Factorise 4x^2 - 12x. Step 1: Find the greatest common factor (GCF) of the terms. The GCF of 4x^2 and 12x is 4x. Step 2: Factor out the GCF. 4x(x - 3) The factorised expression is 4x(x - 3). Question 6.3.2 Factorise x^2 + 4x + 3. Step 1: Find two numbers that multiply to 3 and add to 4. These numbers are 1 and 3. Step 2: Write the quadratic as a product of two binomials. (x+1)(x+3) The factorised expression is (x+1)(x+3). Question 6.3.3 Factorise x^2 - 4. Step 1: Recognize this as a difference of squares, a^2 - b^2 = (a-b)(a+b). x^2 - 2^2 Step 2: Apply the difference of squares formula. (x-2)(x+2) The factorised expression is (x-2)(x+2). Question 7.1.1 What are missing values of x and y on a flow diagram? The rule is y = x-2. Step 1: For the missing output y when input x=2: y = 2 - 2 y = 0 Step 2: For the missing input x when output y=-1: -1 = x - 2 -1 + 2 = x x = 1 The missing values are x=1 and y=0. Question 7.1.2 Complete the table below. The rule is y = x-2. Step 1: Apply the rule y=x-2 for each given x value to find the corresponding y value. For x = -1: y = -1 - 2 = -3 For x = 0: y = 0 - 2 = -2 For x = 2: y = 2 - 2 = 0 For x = 4: y = 4 - 2 = 2 Step 2: The completed table is: |c|c|c|c|c| x & -1 & 0 & 2 & 4 \\ y = x-2 & -3 & -2 & 0 & 2 \\ The completed values for the table are y=2 for x=4. Question 7.1.3 Describe the rule that is used to find the output of the function plane. The rule is y = x-2. This means to find the output, you subtract 2 from the input value. Question 6.4.1 Solve x+2=4. Step 1: Subtract 2 from both sides of the equation. x + 2 - 2 = 4 - 2 Step 2: Simplify the equation. x = 2 The solution is x=2. Question 6.4.2 Solve (x-5)(x+2)=0. Step 1: Set each factor equal to zero. x-5 = 0 or x+2 = 0 Step 2: Solve each linear equation for x. x = 5 or x = -2 The solutions are x=5 or x=-2. Question 6.4.3 Solve x^2-25=0. Step 1: Add 25 to both sides of the equation. x^2 = 25 Step 2: Take the square root of both sides, remembering both positive and negative roots. x = ±sqrt(25) x = ± 5 The solutions are x=5 or x=-5. Send me the next one 📸