Step-by-step explanation:
[tex]x^{2} - 4x - 7 = 0[/tex]
First, let's move the [tex]7[/tex] to the right-hand side so we can determine what constant we'll need on the left-hand side to complete the square:
[tex]x^{2} - 4x = 7[/tex]
From here, since the coefficient of the [tex]x[/tex] term is [tex]-4[/tex], we know the square will be [tex](x - 2)[/tex] (since [tex]-2[/tex] it's half of [tex]-4[/tex]).
To complete this square, we will need to add [tex](-2)^{2}[/tex] to both sides of the equation:
[tex]x^{2} - 4x + (-2)^2 = 7 + ^{-2}[/tex]
[tex]x^{2} - 4x + 4 = 7 + 4[/tex]
[tex](x - 2)^{2} = 11[/tex]
Now we can take the square root of both sides to figure out the solutions to [tex]x[/tex]:
[tex]x - 2 = \pm \sqrt{11}[/tex]
[tex]x = 2 \pm \sqrt{11}[/tex]
