For each equation, (a) solve for x in terms of y.. See Example 8....

Question

For each equation, (a) solve for x in terms of y.. See Example 8. 4x^2 - 2xy + 3y^2 = 2

Accepted Answer

Hey, everyone here we are asked to express X in terms of Y for the following equation, 13 X squared plus eight X Y minus 10, Y squared is equal to six. Now to start solving this problem, we first want to equate our given equation to zero. So that means we just need to move this positive six to the left side of our equation. So doing this, we have 13 X squared plus eight X Y minus 10 Y squared minus six is equal to zero. So now looking at this equation that we have so far, we see that it follows the form of a quadratic equation which is A X squared plus B X plus C is equal to zero. So in our equation, since we are expressing the variable X, in terms of Y, we need to treat Y as a constant. So now we just want to identify our values for A B and C and starting with the value for A. Well, A is the coefficient of X squared. So A is just going to be 13. And now our value for B well, B is just the coefficient of X which we see in our case is A Y. And lastly, our value for C will be the remaining constant in our equation, which in this case is negative 10 Y squared minus six. So now that we have identified our values for A B and C, we can apply, apply the quadratic formula. Well, first we need to recall the quadratic formula which is X is equal to negative B plus or minus the square root of B squared minus four A C all divided by two A. So now we have already identified our values for A B and C. So all we need to do is substitute them into our formula. So doing this, we can solve for X and so we have X is equal to negative eight Y plus or minus the square root of the quantity of eight Y squared minus four times 13 times the quantity of negative 10 Y squared minus six. And again, this is all divided by two times A which is just two times 13. So now we just need to simplify our expression here. So we have negative eight Y plus or minus the square root of and now eight Y squared gives us 64 Y squared minus four times 13, gives us minus 52 times the quantity of negative 10 Y squared minus six. And once again, this is all divided by two times 13, which simplifies to just 26. So now we just need to continue to simplify our expression by next, just factoring in this negative 52 that's underneath the root. So doing this, we have negative eight Y plus or minus the square root of 64 Y squared minus sorry plus because negative 52 times negative 10 Y squared gives us plus Y squared. And then we have plus 312. And this again is all divided by 26. And now just combining the like terms underneath the root, we find our simplified answer is that X is equal to negative eight Y plus or minus the square root of 64 plus 520 gives us 584 Y squared plus 312 all divided by 20 C six. And with this, we have solved our original equation as X in terms of Y. And so we have found our final solution here which is answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

For each equation, (a) solve for x in terms of y.. See Example 8. 4x^2 - 2xy + 3y^2 = 2

Key Concepts

Quadratic Equations

Solving for a Variable

Discriminant

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