Question

When deriving the quadratic formula by completing the square, what expression can be added to both sides of the equation to create a perfect square trinomial?

Answer 1

Answer:

According to steps 2 and 4. The second-order polynomial must be added by $-c$ and $b^{2}$ to create a perfect square trinomial.

Step-by-step explanation:

Let consider a second-order polynomial of the form $a\cdot x^{2} + b\cdot x + c = 0$, $\forall \,x \in\mathbb{R}$. The procedure is presented below:

1) $a\cdot x^{2} + b\cdot x + c = 0$ (Given)

2) $a\cdot x^{2} + b \cdot x = -c$ (Compatibility with addition/Existence of additive inverse/Modulative property)

3) $4\cdot a^{2}\cdot x^{2} + 4\cdot a \cdot b \cdot x = -4\cdot a \cdot c$ (Compatibility with multiplication)

4) $4\cdot a^{2}\cdot x^{2} + 4\cdot a \cdot b \cdot x + b^{2} = b^{2}-4\cdot a \cdot c$ (Compatibility with addition/Existence of additive inverse/Modulative property)

5) $(2\cdot a \cdot x + b)^{2} = b^{2}-4\cdot a \cdot c$ (Perfect square trinomial)

According to steps 2 and 4. The second-order polynomial must be added by $-c$ and $b^{2}$ to create a perfect square trinomial.

Answer 2

it can also be explained as D on edg if you still needed the answer :)