Answer:
The given expression [tex]x^2 + 6\, x + 4[/tex] is equivalent to [tex](x + 3)^2 - 5[/tex]. In this expression, [tex]a = 3[/tex] whereas [tex]b = -5[/tex].
Step-by-step explanation:
Expand [tex](x + a)^2 + b[/tex] using binomial expansion.
[tex]\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}[/tex].
Compare this expression to [tex]x^2 + 6\, x + 4[/tex] to find information about [tex]a[/tex] and [tex]b[/tex].
In particular, these two expressions are supposed to be equal to one another. Therefore:
- The coefficient of the [tex]x^2[/tex] term in these two expressions should be the same. The coefficient of [tex]x^2\![/tex] in both expression is [tex]1[/tex]. That does not provide any information about [tex]a[/tex] or about [tex]b[/tex].
- The coefficient of the [tex]x[/tex] term in these two expressions should be the same. In the first equation, the coefficient of [tex]x\![/tex] is [tex]2\, a[/tex]. In the second equation, that coefficient is [tex]6[/tex]. Therefore, [tex]2\, a = 6[/tex].
- The constant term of these two expressions should be the same. That gives the equation: [tex]a^2 + b = 4[/tex].
The first equation [tex]2\, a = 6[/tex] implies that [tex]a = 3[/tex]. Substitute that value into the second equation and solve for [tex]b[/tex]. The conclusion is that [tex]a= 3[/tex] and [tex]b = -5[/tex].
Therefore, the original equation is equivalent to [tex](x + 3)^2 - 5[/tex].
