Question

Lea solves the equation below by first squaring both sides of the equation.3+2y= √−y. What extraneous solution does Lea obtain?

A) y=− 1/2
B) y=−3/2
C) y=1
D) y=2

Answer 1

The student solved the equation by squaring both sides, leading to a quadratic equation which was factored to obtain two solutions. Upon checking, y = -9/4 was identified as an extraneous solution because it results in an undefined square root. However, this solution does not match the options provided.

Step-by-step explanation:

The equation given is 3 + 2y = √( -y ). When we square both sides, we obtain (3 + 2y)2 = (-y). Let's solve this step by step:

1. Square both sides of the equation: (3 + 2y)2 = (-y)2
2. Expand the left side: 9 + 12y + 4y2 = y
3. Bring all terms to one side: 4y2 + 12y + 9 - y = 0
4. Simplify the equation: 4y2 + 11y + 9 = 0
5. Factor the quadratic equation: (4y + 9)(y + 1) = 0
6. Find the roots by setting each factor equal to zero: 4y + 9 = 0 or y + 1 = 0
7. Solve for y: y = -9/4 or y = -1

Now, we must check these solutions in the original equation. When we substitute y = -1, we have 3 + 2(-1) = √( -(-1) ), which is a valid solution. However, substituting y = -9/4 leads to an undefined square root (because the square root of a negative number is not real), which makes it an extraneous solution. Thus, the extraneous solution that Lea obtained is y = -9/4, which is not one of the options provided. Therefore, there might have been an error in squaring the equation or in the proposed options since none of the options equals -9/4. Since we cannot provide an extraneous solution from the options given, it is important to revisit the problem or the options.