# How do you solve x^2 + 2x = 3 by completing the square?

Mar 26, 2017

Because the sign of the x term is positive, we use the pattern:
${\left(x + a\right)}^{2} = {x}^{2} + 2 a x + {a}^{2} \text{ [1]}$

#### Explanation:

Given: ${x}^{2} + 2 x = 3$

Add ${a}^{2}$ to both sides:

${x}^{2} + 2 x + {a}^{2} = 3 + {a}^{2} \text{ [2]}$

Please observe that the left side of equation [2] now resembles the right side of equation [1]. This means that we can set the middle term of equation [1] equal to the middle term of equation [2], to find the value of "a":

$2 a x = 2 x$

$a = 1$

Substitute 1 for "a" in equation [2]:

${x}^{2} + 2 x + {1}^{2} = 3 + {1}^{2} \text{ [3]}$

Because we have completed the square, the left side of equation [3] collapses into a square with $a = 1$ and the right side becomes a single constant:

${\left(x + 1\right)}^{2} = 4 \text{ [4]}$

Perform the square root operation on both sides:

$x + 1 = \pm 2 \text{ [5]}$

Subtract 1 from both sides:

$x = - 1 \pm 2 \text{ [6]}$

$x = 1 \mathmr{and} x = - 3$