Objective: to derive the formula that can be used to solve the whole class of quadratic equations ax2 + bx + c = 0.
Although a specific quadratic equation was used to introduce the method Solution by Completing the Square, the approach used there actually makes it possible for us to treat a whole class of equations ax2 + bx + c = 0 by means of literal coefficients. In this section, we will derive the formula that we can use to solve the entire class of quadratic equations.
First, divide both sides by the coefficient a (aSquare root of A0, because a quadratic equation is assumed):
Next, add the right expression that can help transfer the first two terms into a square form to both sides:
Rearranging the terms, we have:
Combining the first three terms into a square, we have:
Now, move the right term to the left side:
Making the second main expression into a square:
Simplifying the second major term, we obtain:
Changing the equation into factored form, we get:
Therefore, the two solutions are:
Normally, we write two solutions together, as follows:
source:
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