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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Hey SAMSON🥰, good to see you again. 32 Factorize: 3p^2 + 16 This expression is a sum of two terms, where both terms are positive for any real value of p. It cannot be factorized into simpler expressions with real coefficients. 33 A straight line AB passes through points A(3,2) and B(5,6). Calculate the gradient of the line. Step 1: Identify the coordinates of the points. Let (x_1, y_1) = (3,2) and (x_2, y_2) = (5,6). Step 2: Use the gradient formula. The gradient m is given by the formula: m = (y_2 - y_1)/(x_2 - x_1) Step 3: Substitute the values and calculate. m = (6 - 2)/(5 - 3) m = (4)/(2) m = 2 The gradient of the line is 2. 34 A straight line whose gradient is (1)/(3) passes through points R(8,4) and P(5,k). Find the value of k. Step 1: Identify the given gradient and points. Given gradient m = (1)/(3). Let (x_1, y_1) = (8,4) and (x_2, y_2) = (5,k). Step 2: Use the gradient formula. m = (y_2 - y_1)/(x_2 - x_1) Step 3: Substitute the known values into the formula. (1)/(3) = (k - 4)/(5 - 8) (1)/(3) = (k - 4)/(-3) Step 4: Solve for k. Multiply both sides by -3: -3 × (1)/(3) = k - 4 -1 = k - 4 Add 4 to both sides: -1 + 4 = k 3 = k The value of k is 3. 35 The height of a flag post must be 8m or more but shorter than 12m. The flag post at Umoja Junior School is 2m tall. Form an inequality to represent the information. Let h be the height of the flag post in meters. "8m or more" means h 8. "shorter than 12m" means h < 12. Combining these two conditions, the inequality is: 8 h < 12 36 The equation of a straight line is given as: 2x + 3y = 9. Determine the gradient of the straight line. Step 1: Rearrange the equation into the slope-intercept form y = mx + c. 2x + 3y = 9 Subtract 2x from both sides: 3y = -2x + 9 Divide all terms by 3: y = (-2x)/(3) + (9)/(3) y = -(2)/(3)x + 3 Step 2: Identify the gradient. In the form y = mx + c, m is the gradient. The gradient of the straight line is -(2)/(3). 37 A straight line passes through points N(0,2) and M(8,5). Determine its equation in the form y = mx + c. Step 1: Calculate the gradient m. Let (x_1, y_1) = (0,2) and (x_2, y_2) = (8,5). m = (y_2 - y_1)/(x_2 - x_1) m = (5 - 2)/(8 - 0) m = (3)/(8) Step 2: Determine the y-intercept c. Since point N is (0,2), this means that when x=0, y=2. In the equation y = mx + c, when x=0, y=c. Therefore, c=2. Step 3: Write the equation of the line. Substitute the gradient m = (3)/(8) and the y-intercept c = 2 into the form y = mx + c. y = (3)/(8)x + 2 The equation of the line is y = (3)/(8)x + 2. Send me the next one 📸