A quadratic equation is an equation of the following form ax2 + bx + c = 0, where x is an unknown variable, a, b, and c are constants, and a≠0 (if a = 0 then it will become a linear equation) because the term ax2 is raised to the second degree, it is called the quadratic term. The bx term is the linear term because it has a degree of one. The c term is the constant term because it has a degree of zero. Quadratic equations can be rearranged to be equal to zero. Solutions to the quadratic equation are called roots or zeros.
Quadratics can be solved by factorization method or using the quadratic formula.
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There are many real-world examples that deal with quadratics and parabolas. throwing a ball, shooting a cannon and hitting a golf ball are all examples of situations that can be modelled by quadratic functions.
Now, As we have the idea about the general form of Quadratic equation ax2 + bx + c = 0 and the values of ‘x’ satisfying the equation are called roots of the equation.
Note: when we draw the curve of the quadratic equation, then the curve will always be in U ( parabolic) shape and coefficient of x2 will decide the nature of the curve.
i.e, the curve open in an upward or downward direction.
Note: If a > 0 then the curve will open in an upward direction and if a < 0 then the curve will open in a downward direction.
For example:- graph of y= x2 + 3x - 4 can be drawn as
Here a = 1 > 0 , b = 3 and c = -4
Here, it is clear that the curve will open in an upward direction.
Here the graph cuts the x-axis at two different points. These points and are nothing but the roots of the equation.
**The roots of the quadratic equation may be real, imaginary (complex), equal, and distinct and it is decided by the discriminant (D) of the quadratic equation which can be calculated with the help of the following formula D = b2 – 4ac.
We will talk about the discriminant (D) later.
Assume, the roots are α and β then we can calculate the roots by using the formula given by
- If c and a are equal then the roots are reciprocal to each other.
- If b = 0, then the roots are equal and are opposite in sign.
For example:-
we have to calculate the roots of the quadratic equation : x2 + 3x – 4 = 0
Here, we can simply factorize it, x2 + 3x -4 can be written as (x + 4)( x – 1) = 0
Then, the given equation will be zero only and only if either (x + 4) = 0 or x -1 =0
If x + 4 = 0 then x = -4 and if x – 1 = 0 then x = 1
Hence x = -4 , 1 will be the roots of the equation.
Now, we can also calculate the roots of the equation with the help of Sridharacharya formula i.e,
Here, a = 1 > 0 , b = 3 and c = - 4
Now,
Now,
Hence, we get 1 and -4 as the roots of the quadratic equation.
Here the roots are real and different (1 and -4) , can we directly comment about the nature of roots?
Yes, we can comment about the nature of the roots directly with-out calculating the actual value of the roots.
Now, we will talk about the discriminant (D). why is it so much important.
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Let D denote the discriminant, D = b2 −4ac. Depending on the sign and value of D, nature of the roots would be as follows:
- If D < 0 and |D| is not a perfect square then the roots will be in the form of p + iq and p – iq where p and q are the real and imaginary parts of the complex roots.
- If D < 0 and |D| is a perfect square then the roots will be in the form of p + iq and p - iq where p and q are both rational.
- If D = 0 i.e, b2 – 4ac = 0 that is both roots are real and equal and have the value equals to -b/2a.
- If D > 0 and D is not a perfect square then the roots will be conjugate surds
- If D > 0 and D is a perfect square then the roots are real, rational and unequal
Now again consider the above example :-
we have already know the roots of the equation as 1 and -4.( both are real and different from each other).
Now, we will try to comment about the nature of the roots with the help of the discriminant (D).
a = 1 > 0 , b = 3 and c = - 4
we know D = b2 – 4ac
put the value of a,b and c
D = 32 - 4 x 1x (-4) = 9 + 16 = 25
D > 0
Here, we get discriminant (D) > 0, then we can directly say that whatever the roots will be but the nature of the roots will be real and distinct. This is the actual application of the discriminant formula, and you will be seeing lots of question-related to this concept in your exams.
Signs of the roots: Let P be the product of roots and S be their sum:-
- If P > 0 & S > 0 then both roots are positive
- If P > 0 &S < 0 then both roots are negative
- If P < 0 & S > 0 then the numerically smaller root is negative and the other root is positive
- If P < 0 & S < 0 then numerically larger root is negative and the other root is positive
Minimum and maximum values of ax2 + bx + c = 0:-
- If a > 0 then will be the minimum value of the quadratic equation at x =.
- If a < 0 then will be the maximum value of the quadratic equation at x =
Finding a quadratic equation:
- If roots are given as a and b then the quadratic equation can be written as (x-a)(x-b) = 0 => x2 −(a+b)x + ab = 0
- If sum s and product p of roots are given then the quadratic equation can be written as x2 –sx + p = 0
- If roots are reciprocals of roots of equation ax2 +bx+c = 0, then the quadratic equation is cx2 + bx + a = 0
- If roots are k more than roots of ax2 + bx + c = 0 then the equation is a(x−k)2 + b(x−k) + c = 0
- If roots are k times roots of ax2 + bx + c = 0 then equation is a(x/k)2 + b(x/k) + c = 0
Descartes Rules: if n times sign changes in a polynomial equation then the polynomial equation can have a maximum of n positive roots and to find the maximum possible number of negative roots, find the number of positive roots of f(-x).
Note:- An equation where the highest power is odd must have at least one real root.
Example:- solve for x , -4x² - 7x +12 = 0
Here a = -4 , b = -7 and c = 12
As a = -4 < 0 then it is clear that the curve will open downward.
Now, we will check the nature of the roots of the equation with the help of discriminant (D) = b2 – 4ac = (-7)2 – 4 x (-4) x 12 = 49 + 192 = 241 > 0 , hence D > 0 then the roots will be real and distinct.
Now, we will calculate the roots with the help of the Sridharacharya formula if we are unable to factorize it.
Then,
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