A quadratic equation takes the form ax^{2} + bx + c = 0, representing a seconddegree polynomial equation, where 'a', 'b', and 'c' are constants, and 'x' is the variable. It is essential for 'a' to be nonzero to validate it as a genuine quadratic equation.
The roots or solutions of a quadratic equation are the values of ‘x’ that satisfy the equation, making it true. A quadratic equation can have two real roots, two complex roots, or one real root (in case of a perfect square). The number of roots is determined by the value of the discriminant (Δ) given by:
Δ = b^{2} −4ac
To find the roots of a quadratic equation, several methods can be used:
Rules for Quadratic Equations
Examples
Example1: If x^{2}  3x + 1 = 0, find the value of x + 1/x,
(a) 0
(b) 3
(c) 2
(d) 1
Ans: (b)
Given equation is
x^{2}  3x + 1 = 0 ⇒ x^{2} + 1 = 3x
⇒ x^{2} + 1/x = 3
⇒ x^{2}/x + 1/x = 3
∴ x + 1/x = 3
Example 2: For what value of k, the equation x2 + 2(k  4) x + 2k = 0 has equal roots ?
(a) 6 and 4
(b) 8 and 2
(c) 10 and 4
(d) 12 and 2
Ans: (b)
Given equation is
x^{2} + 2(k  4)x + 2k = 0
On comparing with ax^{2} + bx + c = 0
Here, a = 1, b = 2(k 4), c = 2k
Since, the root are equal, we have D = 0.
b^{2}  4ac = 0
∴ 4(k  4)^{2}  8k = 0
4(k^{2} + 16  8k)  8k = 0
⇒ 4k^{2} + 64  32k  8k = 0
⇒ 4k^{2}  40k + 64 = 0
⇒ k^{2}  10k + 16 = 0
⇒ k^{2}  8k  2k + 16 = 0
⇒ k(k  8) 2 (k  8) = 0
⇒ (k  8) (k  2) = 0
Hence, the value of k 8 or 2.
Example 3: If α and β are the roots of the equation 4x^{2}  19x + 12 = 0, find the equation having the roots 1/α and 1/β
(a) 4x^{2 }+ 19 + 12 = 0
(b) 12x^{2}  19x + 4 = 0
(c) 12x^{2} + 19x + 4 = 0
(d) 4x^{2} + 19x  12 = 0
Ans: (b)
Given equation is 4x^{2}  19x + 12 = 0
Let given equation having the roots 1/α and 1/β,
Then required equuation is
12x^{2}  19x + 4 = 0
Example 4: If one of the roots of quadratic equation 7y2  50y + k = 0 is 7, then what is the value of k ?
(a) 7
(b) 1
(c) 50/7
(d) 7/50
Ans: (a)
Given quadratic equation is
7y^{2}  50y + k = 0
If one root is 7, then it will satisfy the equation i.e putting y = 7 in equation
7 x (7)^{2}  50 x 7 + k = 0
⇒ 7 x 49  350 + k = 0
⇒ 343  350 + k = 0
∴ k = 7
Example 5: The quadrictic equation whose roots are 3 and 1, is
(a) x^{2}  4x + 3 = 0
(b) x^{2}  2x  3 = 0
(c) x^{2} + 2x  3 = 0
(d) x^{2} + 4x + 3 = 0
Ans: (b)
Given that, the roots of the quadrictic equation are 3 and 1.
Let α = 3 and β = 1
Sum of roots = α + β = 3  1 = 2
Products of roots = α . β = (3) (1) = 3
∴ Required quadric equation is
x^{2}  (α + β)x + α β = 0
⇒ x^{2}  (2)x + (3) = 0
⇒ x^{2}  2x  3 = 0
314 videos170 docs185 tests

1. What is a quadratic equation? 
2. How do you solve a quadratic equation by factoring? 
3. Can a quadratic equation have more than two solutions? 
4. What is the quadratic formula and how is it used to solve quadratic equations? 
5. Are there any reallife applications of quadratic equations? 
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