Introduction To Quadratic Equation


Quadratic Equation / Saturday, December 15th, 2018
(Last Updated On: December 15, 2018)

Quadratic Equation Definition

An equation of the form ax2 + 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 = b2 – 4ac is known as the discriminant of the equation ax2 + bx + c = 0 and it’s roots are

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

Relation between Roots and Coefficients

Let ax2 + 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 ax2 + 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) = ax2 + bx + c or y = ax2 + bx + c, then there are six possible graphs

A) When a > 0 and D > 0

i) Y > 0, when X < X1 and X > X2 i.e., graph of the curve is alone X-axis

ii) Y < 0, when X ϵ (X1, X2)

iii) Y = 0, when X =( X1 , X2)

Quadratic Equation 01

B) When a > 0 and D = 0

i) Y ≥ 0 for all x ϵ R

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

Quadratic Equation 02

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

Quadratic Equation 03

D) When a < 0 and D > 0

i) Y > 0, when X ϵ (X1, X2)

ii) Y = 0, when X = (X1, X2)

iii) Y < 0, when X < X1 or X > X2

Quadratic Equation 04

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

Quadratic Equation 05

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

Quadratic Equation 06

Transformation of Quadratic equation

If ax2 + 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 ax2 + 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 x1/2.

Rule 6. Roots are cube roots of given equation, replace x by x1/3.

Nature of Roots

Consider a quadratic equation ax2 + 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 = b2 – 4ac

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

 

 

 

 

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