Answer :
To solve the equation [tex]\( x^2 + 4x = 1 \)[/tex] by completing the square, follow these steps:
1. Rewrite the equation in standard form:
[tex]\[ x^2 + 4x - 1 = 0 \][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[ x^2 + 4x = 1 \][/tex]
3. Complete the square:
To complete the square, add and subtract the square of half the coefficient of [tex]\( x \)[/tex] within the equation. The coefficient of [tex]\( x \)[/tex] is 4, so half of this coefficient is 2 and its square is [tex]\( 2^2 = 4 \)[/tex].
[tex]\[ x^2 + 4x + 4 - 4 = 1 \][/tex]
This can be written as:
[tex]\[ (x + 2)^2 - 4 = 1 \][/tex]
4. Isolate the perfect square term:
[tex]\[ (x + 2)^2 = 1 + 4 \][/tex]
[tex]\[ (x + 2)^2 = 5 \][/tex]
5. Take the square root of both sides:
[tex]\[ x + 2 = \pm \sqrt{5} \][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -2 \pm \sqrt{5} \][/tex]
Hence, the solutions are:
[tex]\[ x = -2 + \sqrt{5} \][/tex]
[tex]\[ x = -2 - \sqrt{5} \][/tex]
Given the options, the correct choice is:
[tex]\[ x = \pm \sqrt{5} - 2 \][/tex]
1. Rewrite the equation in standard form:
[tex]\[ x^2 + 4x - 1 = 0 \][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[ x^2 + 4x = 1 \][/tex]
3. Complete the square:
To complete the square, add and subtract the square of half the coefficient of [tex]\( x \)[/tex] within the equation. The coefficient of [tex]\( x \)[/tex] is 4, so half of this coefficient is 2 and its square is [tex]\( 2^2 = 4 \)[/tex].
[tex]\[ x^2 + 4x + 4 - 4 = 1 \][/tex]
This can be written as:
[tex]\[ (x + 2)^2 - 4 = 1 \][/tex]
4. Isolate the perfect square term:
[tex]\[ (x + 2)^2 = 1 + 4 \][/tex]
[tex]\[ (x + 2)^2 = 5 \][/tex]
5. Take the square root of both sides:
[tex]\[ x + 2 = \pm \sqrt{5} \][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -2 \pm \sqrt{5} \][/tex]
Hence, the solutions are:
[tex]\[ x = -2 + \sqrt{5} \][/tex]
[tex]\[ x = -2 - \sqrt{5} \][/tex]
Given the options, the correct choice is:
[tex]\[ x = \pm \sqrt{5} - 2 \][/tex]
