Question

найдите все значения а, при которых ax^2 +(5a+22)x+9a имеет один корень!

Answer 1

 надеюсь поймешь)

a=22 и a=-2

 

Answer 2

Finding the Values of 'a' for which the Quadratic Equation has One Root

To find the values of 'a' for which the quadratic equation ax^2 + (5a+22)x + 9a has one root, we can use the discriminant of the quadratic equation. The discriminant is given by the formula b^2 - 4ac, where 'a', 'b', and 'c' are the coefficients of the quadratic equation.

If the discriminant is equal to zero, then the quadratic equation has one root. Therefore, we need to find the values of 'a' for which the discriminant is zero.

Let's calculate the discriminant and solve for 'a':

1. Calculate the discriminant: - The coefficient of 'x^2' is 'a'. - The coefficient of 'x' is '5a+22'. - The constant term is '9a'.

The discriminant is given by: (5a+22)^2 - 4(a)(9a).

2. Simplify the discriminant: - Expand the square: (25a^2 + 220a + 484) - 36a^2. - Combine like terms: -11a^2 + 220a + 484.

3. Set the discriminant equal to zero and solve for 'a': - -11a^2 + 220a + 484 = 0.

To find the values of 'a' that satisfy this equation, we can use the quadratic formula:

a = (-b ± √(b^2 - 4ac)) / (2a).

In this case, 'a' is equal to -11, 'b' is equal to 220, and 'c' is equal to 484.

4. Solve for 'a': - Substitute the values into the quadratic formula: a = (-220 ± √(220^2 - 4(-11)(484))) / (2(-11)).

- Simplify the equation: a = (-220 ± √(48400 + 21296)) / (-22).

- Further simplify: a = (-220 ± √69696) / (-22).

- Calculate the square root: a = (-220 ± 264) / (-22).

- Simplify the equation: a = (-220 + 264) / (-22) or a = (-220 - 264) / (-22).

- Calculate the values of 'a': a = 44 / (-22) or a = -484 / (-22).

- Simplify the fractions: a = -2 or a = 22.

Therefore, the values of 'a' for which the quadratic equation ax^2 + (5a+22)x + 9a has one root are a = -2 and a = 22.