Question

Пусть x=2-корень уравнения 2x^2+px-2=0.Найти p и разложить левую часть уравнения на множители.заранее спасибо:)

Answer 1

Если х=2 - корень уравнения, то при подстановке числа 2 вместо х, будем получать верное равенство.
$2\cdot 4+p\cdot 2-2=0,\;\;\;2p=-6,\;\;p=-3\\2x^2-3x-2=0\\D=9+16=25\\x_1=\frac{3-5}{4}=-\frac{1}{2},\;\;x_2=2\\2x^2-3x-2=2(x-2)(x+\frac{1}{2})$

Answer 2

Problem Statement

We are given the equation 2x^2 + px - 2 = 0, and we need to find the value of p and factorize the left-hand side of the equation.

Finding the Value of p

To find the value of p, we can use the given equation and substitute x = 2 - √(2x^2 + px - 2). Let's solve for p:

2x^2 + px - 2 = 0

Substituting x = 2 - √(2x^2 + px - 2):

2(2 - √(2x^2 + px - 2))^2 + p(2 - √(2x^2 + px - 2)) - 2 = 0

Simplifying the equation:

2(4 - 4√(2x^2 + px - 2) + 2x^2 + px - 2) + 2p - p√(2x^2 + px - 2) - 2 = 0

Expanding and rearranging the terms:

8 - 8√(2x^2 + px - 2) + 4x^2 + 2px - 4 + 2p - p√(2x^2 + px - 2) - 2 = 0

Combining like terms:

4x^2 + (2p - 8 - p)√(2x^2 + px - 2) + (2p - 6) = 0

Comparing the coefficients of the terms on both sides of the equation, we can determine the value of p:

4x^2 = 0 => p = 0

Therefore, the value of p is 0.

Factoring the Left-hand Side of the Equation

To factorize the left-hand side of the equation 2x^2 + px - 2 = 0, we can use the quadratic formula or complete the square. Since the value of p is 0, the equation simplifies to:

2x^2 - 2 = 0

We can factor out a 2 from the equation:

2(x^2 - 1) = 0

Now, we have a difference of squares:

2(x - 1)(x + 1) = 0

Therefore, the left-hand side of the equation 2x^2 + px - 2 = 0 can be factored as 2(x - 1)(x + 1).

Summary

The value of p in the equation 2x^2 + px - 2 = 0 is 0. The left-hand side of the equation can be factored as 2(x - 1)(x + 1).

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