Question

X²+2x-24=0 x²-9x+20=0 10n²-9n+2=0 21y²-2y-3=0

Answer 1

Задача 1 ([1, 2, -24])

$1 \cdot x^2 + 2 \cdot x + -24 = 0$

$D = b^2 - 4 \cdot a \cdot c = 2^2 - 4 \cdot 1 \cdot -24 = 100$

D > 0

$x_{1} = \frac{-b - \sqrt{D}}{2a} = \frac{-2 - \sqrt{100}}{2\cdot1} = \frac{-12.0}{2} = -6.0$

$x_{2} = \frac{-b + \sqrt{D}}{2a} = \frac{-2 + \sqrt{100}}{2\cdot1} = \frac{8.0}{2} = 4.0$

Ответ: [-6.0, 4.0]

Задача 2 ([1, -9, 20])

$1 \cdot x^2 + -9 \cdot x + 20 = 0$

$D = b^2 - 4 \cdot a \cdot c = -9^2 - 4 \cdot 1 \cdot 20 = 1$

D > 0

$x_{1} = \frac{-b - \sqrt{D}}{2a} = \frac{9 - \sqrt{1}}{2\cdot1} = \frac{8.0}{2} = 4.0$

$x_{2} = \frac{-b + \sqrt{D}}{2a} = \frac{9 + \sqrt{1}}{2\cdot1} = \frac{10.0}{2} = 5.0$

Ответ: [4.0, 5.0]

Задача 3 ([10, -9, 2])

$10 \cdot x^2 + -9 \cdot x + 2 = 0$

$D = b^2 - 4 \cdot a \cdot c = -9^2 - 4 \cdot 10 \cdot 2 = 1$

D > 0

$x_{1} = \frac{-b - \sqrt{D}}{2a} = \frac{9 - \sqrt{1}}{2\cdot10} = \frac{8.0}{20} = 0.4$

$x_{2} = \frac{-b + \sqrt{D}}{2a} = \frac{9 + \sqrt{1}}{2\cdot10} = \frac{10.0}{20} = 0.5$

Ответ: [0.4, 0.5]

Задача 4 ([21, -2, -3])

$21 \cdot x^2 + -2 \cdot x + -3 = 0$

$D = b^2 - 4 \cdot a \cdot c = -2^2 - 4 \cdot 21 \cdot -3 = 256$

D > 0

$x_{1} = \frac{-b - \sqrt{D}}{2a} = \frac{2 - \sqrt{256}}{2\cdot21} = \frac{-14.0}{42} = -0.3333333333333333$

$x_{2} = \frac{-b + \sqrt{D}}{2a} = \frac{2 + \sqrt{256}}{2\cdot21} = \frac{18.0}{42} = 0.42857142857142855$

Ответ: [-0.3333333333333333, 0.42857142857142855]

Answer 2

To solve these quadratic equations, you can use the quadratic formula, which is given by:

For an equation of the form: ax^2 + bx + c = 0

The solutions for x are given by:

x = (-b ± √(b² - 4ac)) / (2a)

Let's solve each of the equations you provided:

1. x² + 2x - 24 = 0 a = 1, b = 2, c = -24

   x = (-2 ± √(2² - 4 * 1 * (-24))) / (2 * 1) x = (-2 ± √(4 + 96)) / 2 x = (-2 ± √100) / 2 x = (-2 ± 10) / 2

   The two solutions are: x₁ = (-2 + 10) / 2 = 8 / 2 = 4 x₂ = (-2 - 10) / 2 = -12 / 2 = -6

2. x² - 9x + 20 = 0 a = 1, b = -9, c = 20

   x = (9 ± √((-9)² - 4 * 1 * 20)) / (2 * 1) x = (9 ± √(81 - 80)) / 2 x = (9 ± √1) / 2

   The two solutions are: x₁ = (9 + 1) / 2 = 10 / 2 = 5 x₂ = (9 - 1) / 2 = 8 / 2 = 4

3. 10n² - 9n + 2 = 0 a = 10, b = -9, c = 2

   n = (9 ± √((-9)² - 4 * 10 * 2)) / (2 * 10) n = (9 ± √(81 - 80)) / 20 n = (9 ± √1) / 20

   The two solutions are: n₁ = (9 + 1) / 20 = 10 / 20 = 0.5 n₂ = (9 - 1) / 20 = 8 / 20 = 0.4

4. 21y² - 2y - 3 = 0 a = 21, b = -2, c = -3

   y = (2 ± √((-2)² - 4 * 21 * (-3))) / (2 * 21) y = (2 ± √(4 + 252)) / 42 y = (2 ± √256) / 42

   The two solutions are: y₁ = (2 + 16) / 42 = 18 / 42 = 3/7 y₂ = (2 - 16) / 42 = -14 / 42 = -1/3

So, the solutions to the given quadratic equations are:

1. x₁ = 4, x₂ = -6
2. x₁ = 5, x₂ = 4
3. n₁ = 0.5, n₂ = 0.4
4. y₁ = 3/7, y₂ = -1/3