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Let's solve the math problems step-by-step. ### 1. Решите уравнение **a) \( 5x^2 - 12x = 0 \)** Factor out \( x \): \[ x(5x - 12) = 0 \] This gives: \[ x = 0 \] \[ 5x - 12 = 0 \] \[ 5x = 12 \] \[ x = \frac{12}{5} \] **b) \( x^2 + 4x = 0 \)** Factor out \( x \): \[ x(x + 4) = 0 \] This gives: \[ x = 0 \] \[ x + 4 = 0 \] \[ x = -4 \] **c) \( 3x^2 + 7x + 2 = 0 \)** Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where \( a = 3, b = 7, c = 2 \). Calculate: \[ b^2 - 4ac = 49 - 24 = 25 \] So: \[ x = \frac{-7 \pm \sqrt{25}}{6} \] \[ x = \frac{-7 \pm 5}{6} \] This gives: \[ x_1 = \frac{-2}{6} = -\frac{1}{3} \] \[ x_2 = \frac{-12}{6} = -2 \] **d) \( (2x - 1)(2x + 1) = 3 \)** Expand and set equal to zero: \[ 4x^2 - 1 = 3 \] \[ 4x^2 = 4 \] \[ x^2 = 1 \] \[ x = \pm 1 \] **e) \( x^2 - 8x + 16 = 0 \)** Rewrite as: \[ (x - 4)^2 = 0 \] \[ x = 4 \] ### 2. Найдите сумму и произведение корней \( x^2 + 7x - 4 = 0 \). Sum of roots: \( -b/a \) = \(-7/1 = -7\). Product of roots: \( c/a \) = \(-4/1 = -4\). ### 3. Одна из сторон прямоугольника на 7 см больше другой, а площадь равна 44 см². Найдите периметр прямоугольника. Let \( x \) be one side, then \( x + 7 \) is the other side. Area: \[ x(x + 7) = 44 \] \[ x^2 + 7x - 44 = 0 \] Solving this quadratic equation using the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where \( a = 1, b = 7, c = -44 \). Calculate: \[ b^2 - 4ac = 49 + 176 = 225 \] So: \[ x = \frac{-7 \pm 15}{2} \] This gives: \[ x_1 = 4, x_2 = -11 \] Only positive value \( x = 4 \). Perimeter: \[ 2(x + x + 7) = 2(4 + 11) = 30 \] ### 4. Число 8 - корень квадратного уравнения \( x^2 - bx - 6 = 0 \). Найдите второй корень и значение \( b \). Substitute 8 into the quadratic: \[ 8^2 - 8b - 6 = 0 \] \[ 64 - 8b = 6 \] \[ 8b = 58 \] \[ b = \frac{58}{8} = 7.25 \] Using Vieta’s formulas: Sum of roots: \( 8 + x = 7.25 \) \[ x = -0.75 \] ### 5. Один из корней уравнения \( x^2 + px + 7 = 0 \) равен 9. Найдите другой корень и коэффициент \( p \). By Vieta's formulas: Sum of roots: \( 9 + x = -p \) Product of roots: \( 9x = 7 \) Solving for the second root: \[ x = \frac{7}{9} \] Substitute back: \[ 9 + \frac{7}{9} = -p \] \[ p = -\left(9 + \frac{7}{9}\right) \] \[ p = -\frac{88}{9} \] These solutions summarize the answers to the problems given.