Step-by-step explanation:
1.
[tex] {ax}^{2} + bx + c[/tex]
In order to complete the square, we must be in the form
[tex] {x}^{2} + bx + c[/tex]
So we divide the first equation by a,
[tex] \frac{ {ax}^{2} + bx + c}{a} = \frac{0}{a} [/tex]
We then get
[tex] {x}^{2} + \frac{b}{a} x + \frac{c}{a} = 0[/tex]
Next thing, we move the term without a variable( called a constant) to the another side using basic algebra.
[tex] {x}^{2} + \frac{b}{a} x = - \frac{c}{a} [/tex]
Next, thing we do , we multiply the linear coefficient by 1/2
then square it
[tex] \frac{b}{a} \times \frac{1}{2} = (\frac{b}{2a} ) {}^{2} = \frac{b {}^{2} }{4 {a}^{2} } [/tex]
We add that term to both sides of the quadratic
[tex] {x}^{2} + \frac{b}{a} x + \frac{ {b}^{2} }{4 {a}^{2} } = - \frac{c}{a} + \frac{b {}^{2} }{4 {a}^{2} } [/tex]
On the left side, we factor using the Perfect square trinomial.
[tex](x + \frac{b}{2a} ) {}^{2} = \frac{ - c}{a} + \frac{ {b}^{2} }{4 {a}^{2} } [/tex]
On the right side, let try to combine these fractions.
Multiply the first fraction by 4a.
We get
[tex](x + \frac{b}{2a} ) {}^{2} = \frac{ {b}^{2} - 4ac}{4 {a}^{2} } [/tex]
Take the square root of both sides.
[tex]x + \frac{b}{2a} = \sqrt{ \frac{ {b}^{2} - 4ac}{4 {a}^{2} } } [/tex]
Square root of 4a^2. is 2a.
[tex]x + \frac{b}{2a} = \frac{ \sqrt{ {b}^{2} - 4ac} }{2a} [/tex]
[tex]x = - \frac{b}{2a} ± \frac{ \sqrt{ {b}^{2} - 4ac } }{2a} [/tex]
Which is the quadratic formula. Note:Remember that square root have both Positve and negative answers so that why we have ±.
