Question

Помогите решить уравнения:a)x^2+16x-51=0б)x^2-14x-72=0Через дискриминант. Найдите корни уравнения:а)x^3-49x=0б)7x^2=(2x+1)^2+3x^2-5Буду очень благодарна!!!!!

Answer 1

x²+16x-51=0
D=256+204=(2√115)²
х₁=(-16+2√115)/2=√115-8
х₂=(-16-2√115)/2=-√115-8

7x²-14x-72=0
D=196+2016=(2√553)²
х₁=(14+2√553)/14=1+√553/7
х₂=(14-2√553)/14=1-√553/7
Через дискриминант.
Найдите корни уравнения:
x³-49x=0
х(х²-49)=х(х+7)(х-7)=0         х₁=0        х₂=-7       х₃=7
7x²=(2x+1)²+3x²-5
7х²=4х²+4х+1+3х²-5
4х=4     х=1

Answer 2

Solving the equations using the discriminant:

To solve the given equations using the discriminant, we need to find the roots of the equations. The discriminant is a mathematical term used to determine the nature of the roots of a quadratic equation.

1. Equation: x^2 + 16x - 51 = 0 The quadratic equation is in the form of ax^2 + bx + c = 0, where a = 1, b = 16, and c = -51.

The discriminant (D) is calculated using the formula: D = b^2 - 4ac.

Substituting the values, we have: D = 16^2 - 4(1)(-51).

Calculating the discriminant, we get: D = 256 + 204 = 460.

The nature of the roots can be determined as follows: - If D > 0, the equation has two distinct real roots. - If D = 0, the equation has one real root (a repeated root). - If D < 0, the equation has no real roots (complex roots).

Since D = 460, which is greater than 0, the equation has two distinct real roots.

To find the roots, we can use the quadratic formula: x = (-b ± √D) / (2a).

Substituting the values, we have: x = (-16 ± √460) / (2 * 1).

Simplifying further, we get: x = (-16 ± √460) / 2.

The roots of the equation are: x = (-16 + √460) / 2 and x = (-16 - √460) / 2.

2. Equation: x^2 - 14x - 72 = 0 The quadratic equation is in the form of ax^2 + bx + c = 0, where a = 1, b = -14, and c = -72.

The discriminant (D) is calculated using the formula: D = b^2 - 4ac.

Substituting the values, we have: D = (-14)^2 - 4(1)(-72).

Calculating the discriminant, we get: D = 196 + 288 = 484.

Since D = 484, which is greater than 0, the equation has two distinct real roots.

Using the quadratic formula: x = (-b ± √D) / (2a).

Substituting the values, we have: x = (14 ± √484) / (2 * 1).

Simplifying further, we get: x = (14 ± √484) / 2.

The roots of the equation are: x = (14 + √484) / 2 and x = (14 - √484) / 2.

Solving the equations:

1. Equation: x^3 - 49x = 0 To solve this equation, we can factor out the common factor of x: x(x^2 - 49) = 0.

Now, we have two factors: x = 0 and x^2 - 49 = 0.

Solving the second factor, we get: x^2 - 49 = 0.

This equation can be factored as the difference of squares: (x - 7)(x + 7) = 0.

Therefore, the roots of the equation are: x = 0, x = 7, and x = -7.

2. Equation: 7x^2 = (2x + 1)^2 + 3x^2 - 5 Simplifying the equation, we have: 7x^2 = 4x^2 + 4x + 1 + 3x^2 - 5.

Combining like terms, we get: 7x^2 - 4x^2 - 3x^2 = 4x + 1 - 5.

Simplifying further, we have: 0 = 4x + 1 - 5.

Combining like terms, we get: 0 = 4x - 4.

Adding 4 to both sides of the equation, we have: 4 = 4x.

Dividing both sides of the equation by 4, we get: x = 1.

Therefore, the solution to the equation is x = 1.

I hope this helps! Let me know if you have any further questions.