Berdasarkan rumusan tersebut, pembelajaran kali ini, kita akan membuktikan rumus Rumus Jumlah, Selisih, dan Hasil Kali Akar-akar Persamaan Kuadrat yang sudah kita gunakan pada pembelajaran sebelumnya.
Misalkan $\displaystyle x_{1}= \frac{-b-\sqrt{D}}{2a}$ dan $\displaystyle x_{2}= \frac{-b+\sqrt{D}}{2a}$, maka
Rumus Jumlah
$\displaystyle x_{1}+x_{2}=\frac{-b-\sqrt{D}}{2a}+\frac{-b+\sqrt{D}}{2a}$
$\displaystyle \Rightarrow x_{1}+x_{2}=\frac{(-b-\sqrt{D})+(-b+\sqrt{D})}{2a}$
$\displaystyle \Rightarrow x_{1}+x_{2}=\frac{-b-\sqrt{D}-b+\sqrt{D}}{2a}$
$\displaystyle \Rightarrow x_{1}+x_{2}=\frac{-2b}{2a}=-\frac{b}{a}$
Jadi $\displaystyle x_{1}+x_{2}=\frac{-b}{a}$
Rumus Hasil Kali
$\displaystyle x_{1}\cdot x_{2}=\frac{-b-\sqrt{D}}{2a}\cdot \frac{-b+\sqrt{D}}{2a}$
$\displaystyle \Rightarrow x_{1}\cdot x_{2}=\frac{(-b-\sqrt{D})\cdot(-b+\sqrt{D})}{4a^{2}}$
karena $\displaystyle (a+b)(a-b)=a^{2}-b^{2}$, maka
$\displaystyle x_{1}\cdot x_{2}=\frac{b^{2}-(\sqrt{D})^{2}}{4a^{2}}$
$\displaystyle \Rightarrow x_{1}\cdot x_{2}=\frac{b^{2}-D}{4a^{2}}$
$\displaystyle \Rightarrow x_{1}\cdot x_{2}=\frac{b^{2}-(b^{2}-4ac)}{4a^{2}}$
$\displaystyle \Rightarrow x_{1}\cdot x_{2}=\frac{b^{2}-b^{2}+4ac}{4a^{2}}$
$\displaystyle \Rightarrow x_{1}\cdot x_{2}=\frac{4ac}{4a^{2}}=\frac{c}{a}$
Jadi $\displaystyle x_{1}\cdot x_{2}=\frac{c}{a}$
Rumus Selisih
$\displaystyle \left |x_{1}- x_{2}\right |=\left |\frac{-b-\sqrt{D}}{2a}-\frac{-b+\sqrt{D}}{2a} \right |$
$\displaystyle \Rightarrow \left |x_{1}- x_{2}\right |=\left |\frac{(-b-\sqrt{D})-(-b+\sqrt{D})}{2a} \right |$
$\displaystyle \Rightarrow \left |x_{1}- x_{2}\right |=\left |\frac{-b-\sqrt{D}+b-\sqrt{D}}{2a} \right |$
$\displaystyle \Rightarrow \left |x_{1}- x_{2}\right |=\left |\frac{-2\sqrt{D}}{2a} \right |$
$\displaystyle \Rightarrow \left |x_{1}- x_{2}\right |=\left |\frac{-\sqrt{D}}{a} \right |$
Karena $\displaystyle a\neq 0$ dan $\displaystyle \sqrt{D}\geqslant 0$, maka
Kasus 1 : Jika $\displaystyle a<0$ dan , maka berdasarkan konsep pembagian bilangan negatif dan negatif diperoleh
$\displaystyle \left |x_{1}- x_{2}\right |=\left |\frac{-\sqrt{D}}{a} \right |=\left |\frac{\sqrt{D}}{a} \right |$
Kasus 1 : Jika $\displaystyle a>0$ dan , maka berdasarkan konsep pembagian bilangan negatif dan positif diperoleh
$\displaystyle \left |x_{1}- x_{2}\right |=\left |\frac{-\sqrt{D}}{a} \right |=\left |\frac{\sqrt{D}}{a} \right |$
Jadi $\displaystyle \left |x_{1}- x_{2}\right |=\left |\frac{\sqrt{D}}{a} \right |$