For the given equation (x+2) (x+4) = x2 + 6x + 8, how many values of x satisfies this equation?a)Two values of xb)One value of xc)All value of xd)No value of xCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Given Equation:
(x^2)(x^4) = x^2 - 6x + 8

Simplifying the Equation:
(x^6) = x^2 - 6x + 8

Writing the Equation in Standard Form:
x^6 - x^2 + 6x - 8 = 0

Factoring the Equation:
(x^2 - 2)(x^4 + 2x^3 + 4x^2 - 4x + 4) = 0

Setting each Factor to Zero:
x^2 - 2 = 0 or x^4 + 2x^3 + 4x^2 - 4x + 4 = 0

Solving the First Factor:
x^2 = 2
Taking square root on both sides,
x = ±√2

Solving the Second Factor:
To solve the quartic equation, we can try factoring by grouping or using synthetic division. However, it can be observed that the second factor does not have any real solutions. This can be determined by checking the discriminant of the quartic equation or by observing the graph of the equation.

Discriminant:
The discriminant of the quartic equation (x^4 + 2x^3 + 4x^2 - 4x + 4) is negative, indicating that the equation does not have any real solutions.

Graphical Analysis:
Plotting the graph of the equation (x^4 + 2x^3 + 4x^2 - 4x + 4), we can see that the graph does not intersect the x-axis. Hence, there are no real solutions to this equation.

Final Answer:
Therefore, the given equation has two values of x that satisfy the equation, which are x = √2 and x = -√2.

Answer

If we solve the L.H.S. (Left Hand Side) of the equation, we get the following value.
(x+2) (x+4) = x² + 4x + 2x + 8 = x²+ 6x + 8.
This value is same as the R.H.S. (Right Hand Side).
So, all the values of x satisfy the equality.

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