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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5. The factors of x^2 + 7x + 6 are: Step 1: Identify the quadratic expression. The expression is x^2 + 7x + 6. Step 2: Find two numbers that multiply to the constant term (6) and add up to the coefficient of the middle term (7). The numbers are 1 and 6, because 1 × 6 = 6 and 1 + 6 = 7. Step 3: Write the factored form of the quadratic expression. x^2 + 7x + 6 = (x+1)(x+6) The factors are (x+1) and (x+6). The correct option is A. A) (x+1) and (x+6) 6. The value of sqrt(6) × sqrt(27) is: Step 1: Combine the terms under a single square root. sqrt(6) × sqrt(27) = sqrt(6 × 27) Step 2: Multiply the numbers inside the square root. 6 × 27 = 162 So, the expression becomes sqrt(162). Step 3: Simplify the square root by finding perfect square factors of 162. 162 = 81 × 2 162 = 9^2 × 2 Step 4: Extract the perfect square from the square root. sqrt(162) = sqrt(9^2 × 2) = sqrt(9^2) × sqrt(2) = 9sqrt(2) The value is 9sqrt(2). The correct option is B. B) 9sqrt(2) 7. Observe the triangles, the correct statement among the following is: Step 1: Analyze the given triangles ABC and QPR. In ABC: Side AB = 6 cm Side BC = 4 cm Angle B = 90^ (indicated by the square symbol) In QPR: Side QP = 6 cm Side PR = 4 cm Angle P = 90^ (indicated by the square symbol) Step 2: Compare corresponding sides and angles. We have: AB = QP = 6 cm B = P = 90^ BC = PR = 4 cm Step 3: Apply the Side-Angle-Side (SAS) congruence criterion. Since two sides and the included angle of ABC are equal to two sides and the included angle of QPR, the triangles are congruent. The correspondence of vertices is A Q, B P, C R. Step 4: Write the congruence statement. ABC QPR The correct option is A. A) ABC QPR 8. If one side of an isosceles triangle is 5 cm & its equal sides measure 3 cm each, then half of its perimeter is: Step 1: Identify the lengths of the sides of the isosceles triangle. The equal sides are 3 cm each. The third side is 5 cm. So, the side lengths are 3 cm, 3 cm, and 5 cm. Step 2: Calculate the perimeter of the triangle. Perimeter P = sum of all sides P = 3 cm + 3 cm + 5 cm P = 11 cm Step 3: Calculate half of the perimeter. Half of perimeter = (P)/(2) = 11 cm2 Half of perimeter = 5.5 cm The correct option is B. B) 5.5 cm 9. Two angles measure (55^ + 3a) and (115^ - 2a) if each angle is a supplement of the other, then calculate the value of "a". Step 1: Understand the definition of supplementary angles. If two angles are supplementary, their sum is 180^. Step 2: Set up the equation using the given angle measures. (55^ + 3a) + (115^ - 2a) = 180^ Step 3: Combine like terms. 55^ + 115^ + 3a - 2a = 180^ 170^ + a = 180^ Step 4: Solve for a. a = 180^ - 170^ a = 10^ The value of a is 10^. a = 10^ 10. Write any two of Euclid's axioms. Here are two of Euclid's axioms: • Things which are equal to the same thing are equal to one another. • The whole is greater than the part.