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: