Complete the square of the terms inside the square brackets:
ax2 + bx + c = a [x2 + ( b/a) x + c/a]
ax2+bx+c=a[x2+(ba)x+ca]ax^2+bx+c=a\Big[x^2+\Big(\dfrac{b}{a}\Big)x+\dfrac{c}{a}\Big]ax2+bx+c=a[x2+(ab)x+ac]
=a[x2+(ba)x+ca+b24a2−b24a2]=a\Big[x^2+\Big(\dfrac{b}{a}\Big)x+\dfrac{c}{a}+\dfrac{b^2}{4a^2}-\dfrac{b^2}{4a^2}\Big]=a[x2+(ab)x+ac+4a2b2−4a2b2]
=a[(x2+(ba)x+b24a2)−b24a2+ca]=a\Big[\Big(x^2+\Big(\dfrac{b}{a}\Big)x+\dfrac{b^2}{4a^2}\Big)-\dfrac{b^2}{4a^2}+\dfrac{c}{a}\Big]=a[(x2+(ab)x+4a2b2)−4a2b2+ac]
=a[(x+b2a)2−b24a2+ca]=a\Big[\Big(x+\dfrac{b}{2a}\Big)^2-\dfrac{b^2}{4a^2}+\dfrac{c}{a}\Big]=a[(x+2ab)2−4a2b2+ac]
=a[x+b2a]2−b24a+c=a\Big[x+\dfrac{b}{2a}\Big]^2-\dfrac{b^2}{4a}+c=a[x+2ab]2−4ab2+c
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