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The problem at hand is to solve the equation .
Let's begin by factoring the expression:
The given equation is:
We can factor out the common factor of from both terms:
To solve for , we set each factor equal to zero:
Solving for , we have:
Next, consider the second factor:
Solving for , we have:
Since has no real solutions, we ignore these solutions in the real number system.
Thus, the only real solution to the equation is:
Solve the following equation:
\( 2x^2-8=x^2+4 \)
Great question! When we solve , we get . Since no real number squared gives a negative result, these are called imaginary solutions and aren't part of the real number system we typically work with.
Look for terms that share the same variable or number! In , both terms contain , so we can factor it out. This is like factoring out 3 from 3x + 3y = 3(x + y).
You'd be missing important information! Even though has no real solutions, recognizing this is part of the complete solution process. Always solve each factor to show your full mathematical reasoning.
Absolutely! The number of real solutions depends on the specific equation. Some polynomials have multiple real solutions, others have just one (like this problem), and some have no real solutions at all.
Factoring reveals the structure of the equation! It shows us exactly why x = 0 is a solution and helps us systematically find all possible solutions. Other methods like graphing might work, but factoring gives us the complete mathematical picture.
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