Roots of a quadratic equations — lesson. Mathematics State Board, Class 10.

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Quadratic equation

A quadratic equation in the variable \(x\) is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are numbers, \(a \ne 0\). The degree of the quadratic equation is \(2\).

Important!

The equation \(ax^2 + bx + c = 0\) is called the standard form of a quadratic equation.

Roots of a quadratic equation

The value of \(x\) that makes the expression \(ax^2 + bx + c\) is zero, called the roots of the quadratic equation.

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

Divide the equation by \(a\).

x^{2} + \frac{b}{a} x + \frac{c}{a} = 0

Move the constant to the right side.

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

Add the square of one-half of the coefficient of \(x\) on both sides.

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

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

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

Taking square root on both sides.

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

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

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

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

Therefore, the roots of \(ax^2 + bx + c = 0\) are

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

and

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

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3. Roots of a quadratic equations

Theory: