Question

what number would ypu add to both sides of x^2+7x=4 to complete the square

Answer 1

For a monic quadratic (factor of 1 on $x^2$) we always add half the coefficient on the linear ($x$) term, so

$\left(\dfrac 7 2\right)^2 = \dfrac{49}{4}$

Answer: 49/4

$x^2 + 7x + \dfrac{49}{4} = 4 + \dfrac{49}{4}$

$(x + \frac 7 2)^2 = 4 + \frac{49}{4}$

Answer 2

$(*)\qquad(a+b)^2=a^2+2ab+b^2\\\\x^2+7x=4\\\\x^2+2\cdot x\cdot3.5=4\ \ \ \ |+3.5^2\\\\\underbrace{x^2+2\cdot x\cdot3.5+3.5^2}_{(*)}=4+3.5^3\\\\(x+3.5)^2=16.25\\\\Answer:\ 3.5^2=12.25$