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 for the questions: 5.1.1 To calculate the value of the discriminant for the quadratic equation 2x^2 - 4x + 7 = 0. Step 1: Identify the coefficients a, b, c. For 2x^2 - 4x + 7 = 0, we have a=2, b=-4, c=7. Step 2: Use the discriminant formula = b^2 - 4ac. = (-4)^2 - 4(2)(7) = 16 - 56 = -40 The value of the discriminant is -40. = -40 5.1.2 To determine the nature of the roots based on the discriminant. Step 1: Analyze the value of the discriminant. From 5.1.1, = -40. Step 2: Conclude the nature of the roots. Since the discriminant < 0, the roots are non-real and unequal. Non-real and unequal roots 5.2 To determine the numerical value(s) of k for which the equation x^2 + 6x = k will have real and equal roots. Step 1: Rewrite the equation in standard quadratic form ax^2+bx+c=0. x^2 + 6x - k = 0 Step 2: Identify the coefficients a, b, c. For x^2 + 6x - k = 0, we have a=1, b=6, c=-k. Step 3: Apply the condition for real and equal roots, which is = b^2 - 4ac = 0. (6)^2 - 4(1)(-k) = 0 36 + 4k = 0 Step 4: Solve for k. 4k = -36 k = (-36)/(4) k = -9 The numerical value of k is -9. k = -9 6.1.1 To simplify 625^(1)/(4) without using a calculator. Step 1: Express the base as a power. We know that 625 = 5 × 5 × 5 × 5 = 5^4. Step 2: Substitute and simplify using exponent rules. 625^(1)/(4) = (5^4)^(1)/(4) = 5^4 × (1)/(4) = 5^1 = 5 The simplified value is 5. 5 6.1.2 To simplify sqrt(2^2x) · 4[3]8^x without using a calculator. Step 1: Simplify the numerator sqrt(2^2x) · 4. sqrt(2^2x) · 4 = sqrt(2^2x) · 2^2 = sqrt(2^2x+2) = (2^2x+2)^(1)/(2) = 2^(1)/(2)(2x+2) = 2^x+1 Step 2: Simplify the denominator [3]8^x. 3]8^x = [3^x = [3]2^3x = (2^3x)^(1)/(3) = 2^(1)/(3)(3x) = 2^x Step 3: Divide the simplified numerator by the simplified denominator. 2^x+12^x = 2^(x+1)-x = 2^1 = 2 The simplified expression is 2. 2 That's 2 down. 3 left today — send the next one.