Quadratic Equations
Category : 10th Class
Quadratic Equations
If p(x) is a quadratic polynomial, then p(x) = 0 is called a quadratic equation.
Note: (i) An equation of degree 2 is called a quadratic equation.
(ii) The quadratic equation of the form \[a{{x}^{2}}+bx+c=0\].
Solution or roots of a quadratic equation: If p(x) = 0 is a quadratic equation, then the zeros of the polynomial p(x) are called the solutions or roots of the quadratic equation p(x)=0.
Note: (i) Since the degree of a quadratic equation is 2, it has 2 roots or solutions.
(ii) x = a is the root of p(x) = 0, if p(a) = 0.
(iii) Finding the roots of a quadratic equation is called solving the quadratic equation.
(a) Factorization method
(i) Splitting the middle term
(ii) Completing the square
(b) Formula method
(a) (i) Splitting the middle term: Consider the quadratic equation\[a{{x}^{2}}+bx+c=0\].
Step 1: Find the product of the coefficient of\[{{x}^{2}}\]and the constant term i.e., ac.
Step 2: (a) If ac is positive, then choose two factors of ac, whose sum is equal to b (the coefficient of the middle term).
(b) If ac is negative, then choose two factors of ac, whose difference is equal to b (the coefficient of the middle term).
Step 3: Express the middle terms as the sum (or difference) of the two factors obtained in step 2. [Now the quadratic equation has 4 terms]
Step 5: Express the given quadratic equation as a product of two binomials, and solve them.
Step 6: The two values obtained in step 5 are the roots of the given quadratic equation.
Note: An important property used in solving a quadratic equation by splitting the middle term.
"If ab = 0, then either a = 0, or b = 0 or both a and b are 0, where 'a' and 'b' are real numbers"
(ii) Completing the square: In some cases where the given quadratic equation can be solved by factorization, a suitable term is added and subtracted. Then terms are regrouped in such a manner that a square is completed by three of the terms. The equation is then solved using factorization method.
Note: Usually, the term added and subtracted is the square of half the coefficient of x.
Note: The roots of the quadratic equation using the quadratic formula are
\[x=\frac{-b\sqrt{{{b}^{2}}-4ac}}{2a}\]and\[x=\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}\]
Nature of roots: For a quadratic equation \[a{{x}^{2}}+bx+c=0,\]\[{{b}^{2}}-4ac\](denoted by D) is called the discriminate.
Value of Discriminate |
Roots |
Nature of roots |
No. of Roost |
Greater than (positive) |
\[\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}\]and \[\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}\] |
Real and distinct |
2 |
Lesser than 0 (Negative) |
\[\frac{-b+i\sqrt{|D|}}{2a}\]and \[\frac{-b-i\sqrt{|D|}}{2a}\] |
Complex or imaginary |
0 |
Equal to 0 |
\[\frac{-b}{2a},\frac{-b}{2a}\] |
Real and |
2 coincident roots or repeated roots |
If \[{{\text{b}}^{\text{2}}}-\text{4ac}>0\], then \[\sqrt{{{b}^{2}}-4ac}\]is a real number and the quadratic equation \[a{{x}^{2}}+bx+c=0\]has two real roots a and p.
If \[{{\text{b}}^{\text{2}}}-\text{4ac}<0\], then \[\sqrt{{{\text{b}}^{\text{2}}}-\text{4ac}}\] is not a real number and the quadratic equation \[a{{x}^{2}}+bx+c=0\] has no real root.
Note: Any root of the quadratic equation that does not satisfy the condition of the problem is discarded.
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