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Answer:
[tex]\alpha + \beta = -\frac{b}{a}[/tex]
[tex]\alpha \beta = \frac{c}{a}[/tex]
Step-by-step explanation:
Given
[tex]ax^2 + bx + c = 0[/tex]
[tex]Roots: \alpha \& \beta[/tex]
Required
Show that:
[tex]\alpha + \beta = -\frac{b}{a}[/tex]
[tex]\alpha \beta = \frac{c}{a}[/tex]
[tex]ax^2 + bx + c = 0[/tex]
Divide through by a
[tex]\frac{a}{a}x^2 + \frac{b}{a}x + \frac{c}{a} = \frac{0}{a}[/tex]
[tex]x^2 + \frac{b}{a}x + \frac{c}{a} = 0[/tex]
The general form of a quadratic equation is:
[tex]x^2 - (Sum)x + (Product) = 0[/tex]
By comparison, we have:
[tex]-(Sum)x = \frac{b}{a}x[/tex]
[tex]-(Sum) = \frac{b}{a}[/tex]
Sum is calculated as:
[tex]Sum = \alpha + \beta[/tex]
So, we have:
[tex]-(\alpha + \beta) = \frac{b}{a}[/tex]
Divide both sides by -1
[tex]\alpha + \beta = -\frac{b}{a}[/tex]
Similarly;
[tex]Product = \frac{c}{a}[/tex]
Product is calculated as:
[tex]Product = \alpha * \beta[/tex]
So, we have:
[tex]\alpha * \beta = \frac{c}{a}[/tex]
[tex]\alpha \beta = \frac{c}{a}[/tex]