Eloise started to solve a radical equation in this way: square root of negative 2x plus 1 − 3 = x square root of negative 2x plus 1 − 3 + 3 = x + 3 square root of negative 2x plus 1 = x + 3 square root of negative 2x plus 1 − 1 = x + 3 − 1 square root of negative 2 x = x + 2 (square root of negative 2 x)2 = (x − 4)2 −2x = x2 − 8x + 16 −2x + 2x = x2 + 8x + 16 + 2x 0 = x2 + 10x + 16 0 = (x + 2)(x + 8) x + 2 = 0 x + 8 = 0 x + 2 − 2 = 0 − 2 x + 8 − 8 = 0 − 8 x = −2 x = −8 Both solutions are extraneous because they don't satisfy the original equation.

Both solutions cannot satisfy the original equation as Eloise makes mathematical error in solving for those solutions. Check Explanation for the breakdown of her steps and the right way to go about solving this.

Step-by-step explanation:

Radical equation is square root of negative 2x plus 1 − 3 = x

In mathematical terms,

√(-2x + 1) - 3 = x

Them her steps,

She adds 3 to both sides,

square root of negative 2x plus 1 − 3 + 3 = x + 3

In mathematical terms,

√(-2x + 1) - 3 + 3 = x + 3

On simplification, she obtains,

square root of negative 2x plus 1 = x + 3

√(-2x + 1) = x + 3

Then, she goes ahead to subtract 1 from both sides,

square root of negative 2x plus 1 − 1 = x + 3 − 1

√(-2x + 1) - 1 = x + 3 - 1

It was at this point that she makes the mistake that scatters everything; she removed the 1 from under square root to take the (- 1) she introduced (without much thoughts). She then obtains

square root of negative 2 x = x + 2

√(-2x) = x + 2 (wrong step, that scatters the process)

She then squares both sides

(square root of negative 2x)² = (x − 4)²

-2x = (x - 4)²

Here, Eloise makes another error, albeit, this is a typographical error. She wrote (x - 4) instead of (x + 2)

From then on, anything she does, whether wrong or right, it will not give the right answer.

−2x = x² − 8x + 16

Then she makes more errors, (-8x) changes to 8x with the introduced 2x on the left hand side doesn't account for this change.

−2x + 2x = x² + 8x + 16 + 2x

0 = x² + 10x + 16

0 = (x + 2)(x + 8)

x+2 = 0 or x+8 = 0

x = -2 or x = -8

Which is evidently wrong.

The correct solving method should have been from the point

√(-2x + 1) = x + 3

Square both sides

-2x + 1 = (x + 3)²

x² + 6x + 9 = - 2x + 1

x² + 8x + 8 = 0

Solving the equation quadratically.

x = - 1.17 or - 6.83

The true value of x that satisfies the given equation that Eloise attempted to solve are -1.17 and -6.83.

Hope this Helps!!!