Question

Докажите, что уравнение не имеет корней: 2) х в квадрате-х+1=0 Методом выделения полного квадрата!

Answer 1

A=1
b=-1
c=1
D=bв квадрате - 4ас=1-4=-3
D<0 корней нет

Answer 2

Solving the Quadratic Equation Using Completing the Square Method

To prove that the equation 2x^2 - x + 1 = 0 has no real roots, we can use the completing the square method.

First, let's rewrite the equation in the standard form ax^2 + bx + c = 0: 2x^2 - x + 1 = 0

We can start by isolating the quadratic terms on one side of the equation: 2x^2 - x = -1

Now, to complete the square, we need to add and subtract the square of half the coefficient of x^2. The coefficient of x^2 is 2, so half of it is 1. The square of 1 is 1, so we add and subtract 1: 2x^2 - x + 1 - 1 = -1

This simplifies to: (2x^2 - x + 1) - 1 = -1

Now, we can rewrite the left-hand side as a perfect square trinomial: 2(x^2 - (1/2)x) + 1 - 1 = -1 2(x^2 - (1/2)x + (1/4)) = -1

Next, we can factor the perfect square trinomial: 2((x - (1/4))^2) = -1

Now, we can solve for x by taking the square root of both sides: x - (1/4) = ±√(-1/2) x - (1/4) = ±i√(1/2)

Since the right-hand side contains the imaginary unit 'i', it indicates that the equation has no real roots. Therefore, the equation 2x^2 - x + 1 = 0 has no real roots.

This conclusion is supported by the nature of the discriminant (D) of the quadratic equation, which is given by the formula D = b^2 - 4ac. In this case, D = (-1)^2 - 4*2*1 = 1 - 8 = -7, which is negative. When the discriminant is negative, the quadratic equation has no real roots.

Therefore, the equation 2x^2 - x + 1 = 0 does not have any real roots.