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Pair of Linear Equations in Two Variables
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Arithmetic Progressions
Quadratic Equations
- Quadratic Equations
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- Solutions of Quadratic Equations by Completing the Square
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- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
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- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
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Introduction to Trigonometry
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notes
In general, a real number α is called a root of the quadratic equation `ax^2 + bx + c = 0, a ≠ 0` if `aα^2 + bα + c = 0`. We also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation. Note that the zeroes of the quadratic polynomial `ax^2 + bx + c=0` and the roots of the quadratic equation `ax^2 + bx + c = 0` are the same.
By substituting arbitrary values for the variable and deciding the roots of quadratic equation is a time consuming process. Let us learn to use factorisation method to find the roots of the given quadratic equation.
`x^2 - 4 x - 5 = (x - 5) (x + 1)`
`(x - 5) and (x + 1)` are two linear factors of quadratic polynomial `x^2 - 4 x - 5.`
So the quadratic equation obtained from `x^2 - 4 x - 5` can be written as `(x - 5) (x + 1) = 0`
`x - 5 = 0 or x + 1 = 0`
∴`x = 5 or x = -1`
∴5 and the -1 are the roots of the given quadratic equation.
While solving the equation first we obtained the linear factors. So we call this method as ’factorization method’ of solving quadratic equation
Ex.(1) `m^2 - 14 m + 13 = 0`
∴`m^2 - 13 m - 1m + 13 = 0`
∴`m (m - 13) -1 (m - 13) = 0`
∴`(m - 13) (m - 1) = 0`
∴`m - 13 = 0 or m - 1 = 0`
∴`m = 13 or m = 1`
∴13 and 1 are the roots of the given quadratic equation.
