# Quadratic Equation Definition

An equation of the form ax^{2} + bx + c = 0, where a, b and c are the set of real numbers and a ≠ 0, is called quadratic equation. Here ‘a’ is called leading coefficient. The quantity D = b^{2} – 4ac is known as the discriminant of the equation ax^{2} + bx + c = 0 and it’s roots are

\[x=\frac{-b\pm \sqrt{D}}{2a}\]

## Relation between Roots and Coefficients

Let ax^{2} + bx + c = 0, a ≠ 0 be quadratic equation having roots α and β then,

\[a{{x}^{2}}+bx+c\equiv a\left( x-\alpha \right)\left( x-\beta \right)\]

\[\Rightarrow a\left[ {{x}^{2}}+\frac{b}{a}x+\frac{c}{a} \right]\equiv a\left[ {{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta \right]\]

\[\text{Sum of roots }=\alpha +\beta =\frac{-b}{a}=-\frac{\text{Coefficient of x}}{\text{Coefficient of }{{\text{x}}^{2}}}\]

\[\text{Product of roots }=\alpha \beta =\frac{c}{a}=\frac{\text{Constant term}}{\text{Coefficient of }{{\text{x}}^{2}}}\]

## Formation of Quadratic equation

A quadratic equation whose roots are α and β is given by

\[\left( x-\alpha \right)\left( x-\beta \right)=0\]

\[\Rightarrow {{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0\]

\[\Rightarrow {{x}^{2}}-\left( \text{sum of roots} \right)x+p\text{roduct of roots}=0\]

## Symmetric roots

If roots of quadratic equation ax^{2} + bx + c = 0, a ≠ 0 be α and β, then

\[(i)\alpha -\beta =\sqrt{{{\left( \alpha +\beta \right)}^{2}}-4\alpha \beta }=\pm \frac{\sqrt{{{b}^{2}}-4ac}}{a}=\pm \frac{\sqrt{D}}{a}\]

\[(ii){{\alpha }^{2}}+{{\beta }^{2}}={{\left( \alpha +\beta \right)}^{2}}-2\alpha \beta =\frac{{{b}^{2}}-2ac}{{{a}^{2}}}\]

\[(iii){{\alpha }^{2}}-{{\beta }^{2}}=\left( \alpha +\beta \right)\sqrt{{{\left( \alpha +\beta \right)}^{2}}-4\alpha \beta }=\pm \frac{b\sqrt{{{b}^{2}}-4ac}}{{{a}^{2}}}\]

\[(iv){{\alpha }^{3}}+{{\beta }^{3}}={{\left( \alpha +\beta \right)}^{3}}-3\alpha \beta \left( \alpha +\beta \right)=-\frac{b\left( {{b}^{2}}-3ac \right)}{{{a}^{3}}}\]

\[(v){{\alpha }^{3}}-{{\beta }^{3}}={{\left( \alpha -\beta \right)}^{3}}+3\alpha \beta \left( \alpha -\beta \right)=\pm \frac{\left( {{b}^{2}}-ac \right)\sqrt{{{b}^{2}}-4ac}}{{{a}^{3}}}\]

\[(vi)\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{{\alpha }^{2}}+{{\beta }^{2}}}{\alpha \beta }=\frac{{{\left( \alpha +\beta \right)}^{2}}-2\alpha \beta }{\alpha \beta }=\frac{{{b}^{2}}-2ac}{ac}\]

\[(vii){{\alpha }^{2}}\beta +\alpha {{\beta }^{2}}=\alpha \beta \left( \alpha +\beta \right)=-\frac{bc}{{{a}^{2}}}\]

## Sign of Quadratic equation and its Graph

Let f(x) = ax^{2} + bx + c or y = ax^{2} + bx + c, then there are six possible graphs

A) When a > 0 and D > 0

i) Y > 0, when X < X_{1} and X > X_{2} i.e., graph of the curve is alone X-axis

ii) Y < 0, when X ϵ (X_{1}, X_{2})

iii) Y = 0, when X =( X_{1} , X_{2})

B) When a > 0 and D = 0

i) Y ≥ 0 for all x ϵ R

ii) Y = 0 at X = -b / 2a

C) When a > 0 and D < 0

i) Graph is always above X-axis i.e, Y > 0, for all x ϵ R

ii) Imaginary roots

D) When a < 0 and D > 0

i) Y > 0, when X ϵ (X_{1}, X_{2})

ii) Y = 0, when X = (X_{1}, X_{2})

iii) Y < 0, when X < X_{1} or X > X_{2}

E) When a < 0 and D = 0

i) Graph is always below the X-axis i.e., Y ≤ 0 for all x ϵ R

ii) Y = 0 at x = -b / 2a

F) When a < 0 and D < 0

i) The graph is neither touching nor intersecting and lies always below the X-axis i.e. Y < 0, for all x ϵ R

ii) Imaginary roots

## Transformation of Quadratic equation

If ax^{2} + bx + c = 0 be a quadratic equation, then following are the rules to transform the quadratic equation

**Rule 1.** To form an equation whose roots are k (≠ 0) times the roots of ax^{2} + bx + c = 0, replace x by x / k.

**Rule 2.** To form an equation whose roots are the negative of the roots in the given equation, replace x by –x.

**Rule 3.** To form an equation whose roots are reciprocal to the roots of the given equation replace x by 1 /x.

**Rule 4.** To form an equation whose roots are k more than the roots of the given equation, replace x by x + k.

**Rule 5.** Roots are square of given equation, replace x by x^{1/2}.

**Rule 6.** Roots are cube roots of given equation, replace x by x^{1/3}.

## Nature of Roots

Consider a quadratic equation ax^{2} + bx + c = 0, where a, b,c ϵ R and a ≠ 0, then

i) D > 0, roots are real and distinct

ii)D = 0, roots are real and equal

iii)D < 0, roots are imaginary, where D = b^{2} – 4ac

iv) D > 0 and perfect square, then the roots are rational and unequal.

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