Test- Ecuatia De Gradul Al Doilea

1. -5 e solutie a ecuatiei 3x²+2x-1=0

Da

Nu

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Explanation

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2. Solutiile ecuatiei 2x²+3x+1=0 sunt -1 si -1/2:

Adevarat

Fals

The statement is true because when we substitute -1 and -1/2 into the equation 2x^2 + 3x + 1 = 0, we get a true statement. This means that both -1 and -1/2 are solutions to the equation.

Explanation

The statement is true because when we substitute -1 and -1/2 into the equation 2x^2 + 3x + 1 = 0, we get a true statement. This means that both -1 and -1/2 are solutions to the equation.

3. Daca discriminantul ecuatiei este negativ, atunci ecuatia are o solutie dubla:

True

False

If the discriminant of an equation is negative, it means that the equation does not have any real solutions. Therefore, it cannot have a double solution. Hence, the statement "ecuatia are o solutie dubla" (the equation has a double solution) is false.

Explanation

If the discriminant of an equation is negative, it means that the equation does not have any real solutions. Therefore, it cannot have a double solution. Hence, the statement "ecuatia are o solutie dubla" (the equation has a double solution) is false.

4. Daca discriminantul ecuatiei este zero, atunci ecuatia nu are solutii:

True

False

If the discriminant of the equation is zero, it means that the equation has one solution. This is because the discriminant is calculated as b^2 - 4ac, where a, b, and c are the coefficients of the equation. If the discriminant is zero, it indicates that the quadratic equation has a double root, which means that it has only one solution. Therefore, the correct answer is False.

Explanation

If the discriminant of the equation is zero, it means that the equation has one solution. This is because the discriminant is calculated as b^2 - 4ac, where a, b, and c are the coefficients of the equation. If the discriminant is zero, it indicates that the quadratic equation has a double root, which means that it has only one solution. Therefore, the correct answer is False.

5. Daca discriminantul ecuatiei este pozitiv, atunci:

ecuatia are 2 solutii reale

ecuatia are o solutie dubla

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6. Daca (x-a)(2x+7)=2x²+x-21, atunci valoarea lui a este:

5

-3

3

-5

To find the value of "a" in the equation (x-a)(2x+7)=2x^2+x-21, we need to expand the left side of the equation and then compare the coefficients of the like terms. Expanding the equation gives us 2x^2 + 7x - 2ax - 7a. Comparing the coefficients of x^2, x, and the constant term on both sides of the equation, we get 2 = 2 (coefficient of x^2), 1 = -2a (coefficient of x), and -21 = -7a (constant term). Solving these equations simultaneously, we find that a = 3.

Explanation

To find the value of "a" in the equation (x-a)(2x+7)=2x^2+x-21, we need to expand the left side of the equation and then compare the coefficients of the like terms. Expanding the equation gives us 2x^2 + 7x - 2ax - 7a. Comparing the coefficients of x^2, x, and the constant term on both sides of the equation, we get 2 = 2 (coefficient of x^2), 1 = -2a (coefficient of x), and -21 = -7a (constant term). Solving these equations simultaneously, we find that a = 3.

7. Daca x₁si x₂ sunt radacinile ecuatiei ax²+bx+c=0, atunci x₁x₂ este:

-c/a

-b/a

B/a

C/a

If x1 and x2 are the roots of the equation ax2+bx+c=0, then x1x2 can be found by multiplying the roots together. In this case, x1x2 is equal to c/a.

Explanation

If x1 and x2 are the roots of the equation ax2+bx+c=0, then x1x2 can be found by multiplying the roots together. In this case, x1x2 is equal to c/a.

8. Daca x₁si x₂ sunt radacinile ecuatiei ax²+bx+c=0, atunci x₁+ x₂ este:

-b/a

B/a

-c/a

C/a

If x1 and x2 are the roots of the equation ax2+bx+c=0, then x1+x2 is equal to -b/a. This can be derived from the fact that the sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term. In this case, the linear term is b and the quadratic term is a, so the sum of the roots is -b/a.

Explanation

If x1 and x2 are the roots of the equation ax2+bx+c=0, then x1+x2 is equal to -b/a. This can be derived from the fact that the sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term. In this case, the linear term is b and the quadratic term is a, so the sum of the roots is -b/a.

9. Descompunerea in factori a trinomului 6x²-5x-1=0 este:

(2x-1)(3x+1)

(6x+1)(x-1)

6(x-1)(x+1/6)

(6x-1)(x+1)

The correct answer is (6x+1)(x-1). This is the correct factorization of the given trinomial 6x2-5x-1=0. The factors are (6x+1) and (x-1). The other option, 6(x-1)(x+1/6), is not a correct factorization of the trinomial.

Explanation

The correct answer is (6x+1)(x-1). This is the correct factorization of the given trinomial 6x2-5x-1=0. The factors are (6x+1) and (x-1). The other option, 6(x-1)(x+1/6), is not a correct factorization of the trinomial.

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10. Ecuatia de gradul al doilea care are radacinile 4 si -3 este:

X2+x-12=0

X2-x-12=0

-x2-x+12=0

-x2+x+12=0

The equation of the second degree that has the roots 4 and -3 can be found using the fact that the sum of the roots is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term, and the product of the roots is equal to the constant term divided by the coefficient of the quadratic term. By applying these formulas, we can determine that the correct equation is x2-x-12=0. The other option, -x2+x+12=0, does not satisfy these conditions.

Explanation

The equation of the second degree that has the roots 4 and -3 can be found using the fact that the sum of the roots is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term, and the product of the roots is equal to the constant term divided by the coefficient of the quadratic term. By applying these formulas, we can determine that the correct equation is x2-x-12=0. The other option, -x2+x+12=0, does not satisfy these conditions.

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