Direction: Find the solution set of the following. 1.x(x+3)=28 2.x(x+6)=8(x+3) 3.(t+1)² + (t-8)²=45 4.(3r+1)² + (r+2)²=65 5.1/x-1 + 1/2=2/x²-1
Answer (1)
Let's solve each equation step by step. 1. x(x + 3) = 28 Rearranging gives:x^2 + 3x - 28 = 0 Now, we can factor or use the quadratic formula. Factoring:(x + 7)(x - 4) = 0 Setting each factor to zero:x + 7 = 0 \quad \Rightarrow \quad x = -7x - 4 = 0 \quad \Rightarrow \quad x = 4 Solution set: x = -7, 4 2. x(x + 6) = 8(x + 3) Expanding both sides gives:x^2 + 6x = 8x + 24 Rearranging gives:x^2 - 2x - 24 = 0 Factoring:(x - 6)(x + 4) = 0 Setting each factor to zero:x - 6 = 0 \quad \Rightarrow \quad x = 6x + 4 = 0 \quad \Rightarrow \quad x = -4 Solution set: x = 6, -4 3. (t + 1)^2 + (t - 8)^2 = 45 Expanding both squares:(t^2 + 2t + 1) + (t^2 - 16t + 64) = 45 Combining like terms:2t^2 - 14t + 65 = 45 Rearranging gives:2t^2 - 14t + 20 = 0 Dividing by 2:t^2 - 7t + 10 = 0 Factoring:(t - 5)(t - 2) = 0 Setting each factor to zero:t - 5 = 0 \quad \Rightarrow \quad t = 5t - 2 = 0 \quad \Rightarrow \quad t = 2 Solution set: t = 5, 2 4. (3r + 1)^2 + (r + 2)^2 = 65 Expanding both squares:(9r^2 + 6r + 1) + (r^2 + 4r + 4) = 65 Combining like terms:10r^2 + 10r + 5 = 65 Rearranging gives:10r^2 + 10r - 60 = 0 Dividing by 10:r^2 + r - 6 = 0 Factoring:(r - 2)(r + 3) = 0 Setting each factor to zero:r - 2 = 0 \quad \Rightarrow \quad r = 2r + 3 = 0 \quad \Rightarrow \quad r = -3 Solution set: r = 2, -3 5. \frac{1}{x - 1} + \frac{1}{2} = \frac{2}{x^2 - 1} First, note that x^2 - 1 = (x - 1)(x + 1) . Rewriting gives:\frac{1}{x - 1} + \frac{1}{2} = \frac{2}{(x - 1)(x + 1)} Multiplying through by 2(x - 1)(x + 1) to eliminate the denominators:2(x + 1) + (x - 1)(x + 1) = 4 Expanding:2x + 2 + x^2 - 1 = 4 Combining like terms:x^2 + 2x + 1 = 4 Rearranging gives:x^2 + 2x - 3 = 0 Factoring:(x + 3)(x - 1) = 0 Setting each factor to zero:x + 3 = 0 \quad \Rightarrow \quad x = -3x - 1 = 0 \quad \Rightarrow \quad x = 1 \quad \text{(not valid, as it makes the denominator zero)} Solution set: x = -3
