Solve the equation for the indicated variable. Assume no denominator is 0.
-$$2 x ^ { 2 } - 4 x y + 3 y ^ { 2 } = 1 , \text { for } x$$
A) $x = \frac { 2 y \pm \sqrt { 2 - 2 y ^ { 2 } } } { 2 }$
B) $x = y \pm ( 1 - y )$
C) $x = - y \pm \sqrt { 1 - y ^ { 2 } }$
D) $x = - y \pm 2 \sqrt { 1 - y ^ { 2 } }$

Question

Quizplus · Accepted Answer

To solve for $x$, we rearrange the equation into a standard quadratic form $ax^2 + bx + c = 0$. The given equation is $2x^2 - 4xy + 3y^2 = 1$. We can rewrite it as $2x^2 - 4xy + (3y^2 - 1) = 0$. Applying the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a = 2$, $b = -4y$, and $c = 3y^2 - 1$, we get $x = \frac{4y \pm \sqrt{(-4y)^2 - 4(2)(3y^2 - 1)}}{2(2)}$. Simplifying inside the square root gives $16y^2 - 8(3y^2 - 1) = 16y^2 - 24y^2 + 8 = -8y^2 + 8$. Thus, $x = \frac{4y \pm \sqrt{-8y^2 + 8}}{4}$, which simplifies to $x = \frac{2y \pm \sqrt{2 - 2y^2}}{2}$, matching choice A.

Solve the Equation for the Indicated Variable 2x2−4xy+3y2=1, for x2 x ^ { 2 } - 4 x y + 3 y ^ { 2 } = 1 , \text { for } x2x2−4xy+3y2=1, for x

Solve the Equation for the Indicated Variable $2 x ^ { 2 } - 4 x y + 3 y ^ { 2 } = 1 , \text { for } x$