Question

Write the following expression in the form (x+a)^2+ba)x^2+6x+4​

Answer 1

Answer:

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

Step-by-step 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 $(x + 3)^2 - 5$.