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 x: 1.1.1 To solve (2x-1)(x-1)=0, we set each factor equal to zero. Step 1: Set each factor to zero. 2x-1=0 or x-1=0 Step 2: Solve for x in each equation. 2x=1 x=(1)/(2) x=1 The solutions are x=(1)/(2) or x=1. x=(1)/(2) or x=1 1.1.2 To solve 2x^2-3x-1=0, we use the quadratic formula x = -b ± sqrt(b^2-4ac)2a. Step 1: Identify the coefficients a, b, c. For 2x^2-3x-1=0, we have a=2, b=-3, c=-1. Step 2: Substitute the values into the quadratic formula. x = -(-3) ± sqrt((-3)^2 - 4(2)(-1))2(2) x = 3 ± sqrt(9 + 8)4 x = 3 ± sqrt(17)4 Step 3: Calculate the two values for x and round to two decimal places. x_1 = 3 + sqrt(17)4 ≈ (3 + 4.1231)/(4) ≈ (7.1231)/(4) ≈ 1.7807 x_2 = 3 - sqrt(17)4 ≈ (3 - 4.1231)/(4) ≈ (-1.1231)/(4) ≈ -0.2807 Rounding to two decimal places, we get x ≈ 1.78 or x ≈ -0.28. x ≈ 1.78 or x ≈ -0.28 1.1.3 To solve the inequality x(2x+1)-3 0, we first expand and rearrange it into standard quadratic form. Step 1: Expand and rearrange the inequality. 2x^2+x-3 0 Step 2: Find the roots of the corresponding quadratic equation 2x^2+x-3=0. We can factor the quadratic expression: (2x+3)(x-1)=0 Setting each factor to zero gives the roots: 2x+3=0 2x=-3 x=-(3)/(2) x-1=0 x=1 The roots are x=-(3)/(2) and x=1. Step 3: Determine the interval where the inequality holds. Since the coefficient of x^2 is positive (2 > 0), the parabola opens upwards. The inequality 2x^2+x-3 0 means we are looking for the values of x where the parabola is below or on the x-axis. This occurs between and including the roots. Therefore, the solution is -(3)/(2) x 1. -(3)/(2) x 1 or -1.5 x 1 Send me the next one 📸