4. Составьте квадратное уравнение, корни которого равны: a) 2 и 5; в) 0 и -3; б) -1 и 0,8; г) 1/2 и -1/4; д) √2 и -√2; е) 1-√2 и 1+√2.

Если известны корни квадратного уравнения \(x_1\) и \(x_2\), то уравнение можно записать в виде: \((x - x_1)(x - x_2) = 0\). Распишем каждое уравнение: а) x₁ = 2, x₂ = 5 \((x - 2)(x - 5) = 0\) => \(x^2 - 5x - 2x + 10 = 0\) => \(x^2 - 7x + 10 = 0\) б) x₁ = -1, x₂ = 0.8 \((x + 1)(x - 0.8) = 0\) => \(x^2 - 0.8x + x - 0.8 = 0\) => \(x^2 + 0.2x - 0.8 = 0\) или \(10x^2 + 2x - 8 = 0\) или \(5x^2 + x - 4 = 0\) в) x₁ = 0, x₂ = -3 \((x - 0)(x + 3) = 0\) => \(x(x + 3) = 0\) => \(x^2 + 3x = 0\) г) x₁ = 1/2, x₂ = -1/4 \((x - 1/2)(x + 1/4) = 0\) => \(x^2 + 1/4x - 1/2x - 1/8 = 0\) => \(x^2 - 1/4x - 1/8 = 0\) или \(8x^2 - 2x - 1 = 0\) д) x₁ = \(\sqrt{2}\), x₂ = -\(\sqrt{2}\) \((x - \sqrt{2})(x + \sqrt{2}) = 0\) => \(x^2 - (\sqrt{2})^2 = 0\) => \(x^2 - 2 = 0\) е) x₁ = \(1 - \sqrt{2}\), x₂ = \(1 + \sqrt{2}\) \((x - (1 - \sqrt{2}))(x - (1 + \sqrt{2})) = 0\) => \((x - 1 + \sqrt{2})(x - 1 - \sqrt{2}) = 0\) => \((x - 1)^2 - (\sqrt{2})^2 = 0\) => \(x^2 - 2x + 1 - 2 = 0\) => \(x^2 - 2x - 1 = 0\) Ответ: a) \(x^2 - 7x + 10 = 0\) б) \(5x^2 + x - 4 = 0\) в) \(x^2 + 3x = 0\) г) \(8x^2 - 2x - 1 = 0\) д) \(x^2 - 2 = 0\) е) \(x^2 - 2x - 1 = 0\)