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| Quadratic Equations |
Quadratic Equations
A quadratic equation is a polynomial function of degree 2, equated to 0 or a constant.
The parent equation for a quadratic function is ax^2+bx+c=0 where x is variable and a, b and c are real constants.
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'a' determines how wide or narrow the function is.
- If |a| is greater than 1, the parabola will be narrow.
- If |a| is less than 1, the parabola will be wider.
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Roots of any function are the values of the parameter(s) where the function equates to zero. Roots of a quadratic equation (function actually) are the value of variable (here it is 'x' since the equation we have taken is quadratic in 'x') which satisfies the equation for a given set of constants (here--> a,b,c).
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Every quadratic equations ax^2+bx+c=0 can be expressed as (x-p)(x-q)=0 where p and q will be the roots of the given quadratic equation. These roots may or may not be real in nature.
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Quadratic functions create a parabola, also known as a 'u' shape.
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The vertex of a quadratic funtion is the turning point of which the graph reflects itself (hence the vertex also relating to 'axis of symmetry', the line in which a quadratic function reflects).
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The values of x where the graph of y=ax^2+bx+c touches the x axis are the roots of the quadratic equation ax^2+bx+c=0.
ROOTS
A quadratic always have 2 roots. In case the quadratic function represents a perfect square, it is said that both the roots have the same value (saying that there is only one root will be wrong since a quadratic equation has to have 2 roots). The nature and value of roots can be calculated using the set of constants associated with it.
Nature of Roots
As stated earlier, the roots of a quadratic equation are not always real. The nature of the roots can be determined easily by calculating the value of D which is given by b^2-4ac
D=b^2-4ac
- If D>0, both the roots will be real in nature.
- If D==0, both the roots will be real and equal in nature.
- If D<0, both the roots will be imaginary in nature (no real value of x will satisfy the equation)
It can be easily observed that that the values of the roots are equal only when D==0 but the nature of roots is always the same for both roots.
Value of Roots
Let the roots of ax^2+bx+c=0 be p and q, then
p= (-b + sqrt(D))/2a
q= (-b - sqrt(D))/2a
- The the equation has imaginary roots, they will always be found in conjugate pairs. For example, if you know that one of roots is 2+3i then you can directly determine the other root as 2-3i by just changing the sign in between the real and imaginary part of value. (This can be inferred from the formula of calculating roots' value.)