"Don't understand the step in Deriving quadratic equationFactorize..." was answered by our experts

High School
Algebra
Algebra I
Quadratic function and equation

Nathanael Hansen

Answered question

2022-04-02

Don't understand the step in Deriving quadratic equation
Factorize the left side:

{(x + \frac{b}{2 a})}^{2} = - \frac{c}{a} + \frac{b^{2}}{4 a^{2}}

Answer & Explanation

Jared Kemp

Beginner2022-04-03Added 14 answers

Step 1
I can't comment due to less reputation. Here is a hint
${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$
Also here is a proof I like using the relationship between the zeros of a polynomial and its constant coefficients. This relationship is for a quadratic polynomial,
$a x^{2} + b x + c$
with zeroes $α$ and $β$
$α β = \frac{c}{a}$
$α + β = \frac{- b}{a}$
Using this,
$(α - β)^{2} & = (α + β)^{2} - 4 α β$
$(α - β)^{2} & = (\frac{- b}{a})^{2} - \frac{4 c}{a}$
$α - β & = \frac{\pm \sqrt{b^{2} - 4 a c}}{a}$
Using this and $α + β = \frac{- b}{a}$ one can easily derive the quadratic formula

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