Question

3x^2+5x+2=0 решите тремя способами (квадратное уравнение)

Answer 1

1. 3х^2+5х+2=0
D=25-4*2*3=1
x=(-5-1)/6=-1
x=(-5+1)/6=-2/3

2. 3х^2+5х+2=0
3x^2+3x+2x+2=0
3x(x+1)+2(x+1)=0
(x+1)(3x+2)=0
x=-1
x=-2/3

Answer 2

Solving the Quadratic Equation 3x^2 + 5x + 2 = 0

To solve the quadratic equation 3x^2 + 5x + 2 = 0 using three different methods, we can use the quadratic formula, factoring, and completing the square.

Using the Quadratic Formula

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by: x = (-b ± √(b^2 - 4ac)) / (2a)

For the given equation 3x^2 + 5x + 2 = 0, where a = 3, b = 5, and c = 2, the solutions for x using the quadratic formula are: x = (-5 ± √(5^2 - 4*3*2)) / (2*3)

Calculating the discriminant (b^2 - 4ac): b^2 - 4ac = 5^2 - 4*3*2 = 25 - 24 = 1

Substituting the values into the quadratic formula: x = (-5 ± √1) / 6 x = (-5 + 1) / 6 = -4 / 6 = -2/3 x = (-5 - 1) / 6 = -6 / 6 = -1

So, the solutions for x using the quadratic formula are: x = -2/3 and x = -1

Using Factoring

To solve the equation 3x^2 + 5x + 2 = 0 by factoring, we need to find two numbers that multiply to give the constant term (2) and add to give the coefficient of the x term (5). These numbers are 2 and 1. Therefore, we can factor the equation as: 3x^2 + 3x + 2x + 2 = 0 3x(x + 1) + 2(x + 1) = 0 (3x + 2)(x + 1) = 0

Setting each factor equal to zero gives us the solutions: 3x + 2 = 0 which gives x = -2/3 x + 1 = 0 which gives x = -1

So, the solutions for x using factoring are: x = -2/3 and x = -1

Using Completing the Square

To solve the equation 3x^2 + 5x + 2 = 0 by completing the square, we first divide through by the coefficient of x^2 to make the coefficient of x^2 equal to 1: x^2 + (5/3)x + 2/3 = 0

Next, we complete the square by adding and subtracting the square of half the coefficient of x: x^2 + (5/3)x + (5/6)^2 - (5/6)^2 + 2/3 = 0 x^2 + (5/3)x + (5/6)^2 = (5/6)^2 - 2/3 x^2 + (5/3)x + (5/6)^2 = 25/36 - 24/36 x^2 + (5/3)x + (5/6)^2 = 1/36

This can be written as: (x + 5/6)^2 = 1/36

Taking the square root of both sides: x + 5/6 = ±√(1/36) x + 5/6 = ±1/6

Solving for x gives us: x = -5/6 ± 1/6 x = -2/3 and x = -1

So, the solutions for x using completing the square are: x = -2/3 and x = -1