Question

which equation has the solutions (x = 1 pm sqrt{5})?

• (x^2 + 2x + 4 = 0)
• (x^2 - 2x + 4 = 0)
• (x^2 + 2x - 4 = 0)
• (x^2 - 2x - 4 = 0)

Explanation:

⚡ Using what you learned: completing the square

Step 1: Isolate the radical term

Start with the given solution:

$$ x = 1 \pm \sqrt{5} $$

Subtract \( 1 \) from both sides to isolate the term with the square root:

$$ x - 1 = \pm\sqrt{5} $$

Step 2: Square both sides

Square both sides of the equation to eliminate the radical:

$$ (x - 1)^2 = (\pm\sqrt{5})^2 $$

$$ x^2 - 2x + 1 = 5 $$

Step 3: Set the equation to zero

Subtract \( 5 \) from both sides to write the quadratic equation in standard form \( ax^2 + bx + c = 0 \):

$$ x^2 - 2x + 1 - 5 = 0 $$

$$ x^2 - 2x - 4 = 0 $$

Answer:

\( x^2 - 2x - 4 = 0 \)