Question

Write the following expression in the form (x+a)^2+b

a)x^2+6x+4​

Answer 1

The given expression
`x^2 + 6\, x + 4` is equivalent to
`(x + 3)^2 - 5`. In this expression,
`a = 3` whereas
`b = -5`.

Explanation:

Expand
`(x + a)^2 + b` using binomial expansion.

`\begin{aligned} & (x + a)^2 + b \\ &= (x + a) \cdot (x + a) + b \\ &= \left(x^2 + a\, x\right) + \left(a\, x + a^2\right) + b \\ &= x^2 + 2\, a\, x + (a^2 + b)\end{aligned}`.

Compare this expression to
`x^2 + 6\, x + 4` to find information about
`a` and
`b`.

In particular, these two expressions are supposed to be equal to one another. Therefore:

• The coefficient of the
  `x^2` term in these two expressions should be the same. The coefficient of
  `x^2\!` in both expression is
  `1`. That does not provide any information about
  `a` or about
  `b`.

• The coefficient of the
  `x` term in these two expressions should be the same. In the first equation, the coefficient of
  `x\!` is
  `2\, a`. In the second equation, that coefficient is
  `6`. Therefore,
  `2\, a = 6`.

• The constant term of these two expressions should be the same. That gives the equation:
  `a^2 + b = 4`.

The first equation
`2\, a = 6` implies that
`a = 3`. Substitute that value into the second equation and solve for
`b`. The conclusion is that
`a= 3` and
`b = -5`.

Therefore, the original equation is equivalent to
`(x + 3)^2 - 5`.