Formation of a quadratic equation — lesson. Mathematics State Board, Class 10.

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Consider the equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).

The roots of the quadratic equation are

\frac{- b + \sqrt{b^{2} - 4 ac}}{2 a}

and

\frac{- b - \sqrt{b^{2} - 4 ac}}{2 a}

If \(\alpha\) and \(\beta\) are the roots of a quadratic equation \(ax^2 + bx + c = 0\), then:

\(\alpha =\)

\frac{- b + \sqrt{b^{2} - 4 ac}}{2 a}

and \(\beta =\)

\frac{- b - \sqrt{b^{2} - 4 ac}}{2 a}

Sum of the roots \(=\) \(\alpha + \beta\)

\(=\)

\frac{- b + \sqrt{b^{2} - 4 ac}}{2 a}

\(+\)

\frac{- b - \sqrt{b^{2} - 4 ac}}{2 a}

\(=\)

\frac{- b + \sqrt{b^{2} - 4 ac} - b - \sqrt{b^{2} - 4 ac}}{2 a}

\(=\)

\frac{- 2 b}{2 a} = \frac{- b}{a}

Sum of the roots \(=\) \(\alpha + \beta\) \(=\)

\frac{- b}{a}

Product of the roots \(=\) \(\alpha \beta\)

\(=\)

\frac{- b + \sqrt{b^{2} - 4 ac}}{2 a}

\(\times\)

\frac{- b - \sqrt{b^{2} - 4 ac}}{2 a}

\(=\)

\frac{(- b) (- b) + (- b) (- \sqrt{b^{2} - 4 ac}) + (\sqrt{b^{2} - 4 ac}) (- b) + (\sqrt{b^{2} - 4 ac}) (- \sqrt{b^{2} - 4 ac})}{2 a \times 2 a}

\(=\)

\frac{b^{2} + b \sqrt{b^{2} - 4 ac} - b \sqrt{b^{2} - 4 ac} - (b^{2} - 4 ac)}{4 a^{2}}

\(=\)

\frac{b^{2} - b^{2} + 4 ac}{4 a^{2}}

\(=\)

\frac{4 ac}{4 a^{2}} = \frac{c}{a}

Product of the roots \(=\) \(\alpha \beta\) \(=\)

\frac{c}{a}

Since \((x - \alpha)\) and \((x - \beta)\) are factors of \(ax^2 + bx + c = 0\):

\((x - \alpha) (x - \beta) = 0\)

\(\Rightarrow x^2 - \alpha x - \beta x + \alpha \beta = 0\)

\(\Rightarrow x^2 - (\alpha + \beta) x + \alpha \beta = 0\)

\(\Rightarrow x^2 - (\text{sum of roots}) x + \text{product of roots} = 0\)

If \(\alpha\) and \(\beta\) are the roots of a quadratic equation, then the general formula to construct the quadratic equation is \(x^2 - (\alpha + \beta) x + \alpha \beta = 0\).

That is, \( x^2 - (\text{sum of roots}) x + \text{product of roots} = 0\).

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4. Formation of a quadratic equation

Theory: