13. Four statements are given below with respect to the nature of roots of the quadratic equation ax² + bx+c = 0. The wrong statement is:
(A) If b² - 4ac > 0, then the roots are real and distinct.
(B) b² - 4ac = 0, then the roots are real and equal.
(C) If b² - 4ac 20, then the roots are imaginary.
(D) If b² - 4ac <then there is no real roots
Answers
Answer:
द्विघात समीकरण के मूलों की प्रकृति पूरी तरह से इसके विवेचक b . के मान पर निर्भर करती है2 - 4एसी।
द्विघात समीकरण में ax2+ bx + c = 0, a ≠ 0 गुणांक a, b और c वास्तविक हैं। हम जानते हैं, समीकरण ax . के मूल (समाधान)2 + bx + c = 0 x = . द्वारा दिया गया है
-ख±
√
ख2-4एसी
2ए
.
1. अगर बी2 - 4ac = 0 तो मूल x = . होंगे
-ख±0
2ए
=
-ख-0
2ए
,
-ख+0
2ए
=
-ख
2ए
,
-ख
2ए
.
स्पष्ट रूप से,
-ख
2ए
एक वास्तविक संख्या है क्योंकि b और a वास्तविक हैं।
अत: समीकरण ax . के मूल2 + bx + c = 0 वास्तविक और बराबर हैं यदि b2 - 4ac = 0। If b2 - 4ac > 0 then b2−4ac−−−−−−−√ will be real and non-zero. As a result, the roots of the equation ax2 + bx + c = 0 will be real and unequal (distinct) if b2 - 4ac > 0.
3. If b2 - 4ac < 0, then b2−4ac−−−−−−−√ will not be real because (b2−4ac−−−−−−−√)2 = b2 - 4ac < 0 and square of a real number always positive.
Thus, the roots of the equation ax2 + bx + c = 0 are not real if b2 - 4ac < 0.
As the value of b2 - 4ac determines the nature of roots (solution), b2 - 4ac is called the discriminant of the quadratic equation.
Definition of discriminant: For the quadratic equation ax2 + bx + c =0, a ≠ 0; the expression b2 - 4ac is called discriminant and is, in general, denoted by the letter ‘D’.
Thus, discriminant D = b2 - 4ac
Note:
Discriminant of
ax2 + bx + c = 0
Nature of roots of
ax2 + bx + c = 0
Value of the roots of
ax2 + bx + c = 0
b2 - 4ac = 0
Real and equal
- b2a, -b2a
b2 - 4ac > 0
Real and unequal
−b±b2−4ac√2a
b2 - 4ac < 0
Not real
No real value
When a quadratic equation has two real and equal roots we say that the equation has only one real solution.
Solved examples to examine the nature of roots of a quadratic equation:
1. Prove that the equation 3x2 + 4x + 6 = 0 has no real roots.
Solution:
Here, a = 3, b = 4, c = 6.
So, the discriminant = b2 - 4ac
= 42 - 4 ∙ 3 ∙ 6 = 36 - 72 = -56 < 0.
Therefore, the roots of the given equation are not real.
2. Find the value of ‘p’, if the roots of the following quadratic equation are equal (p - 3)x2 + 6x + 9 = 0.
Solution:
For the equation (p - 3)x2 + 6x + 9 = 0;
a = p - 3, b = 6 and c = 9.
Since, the roots are equal
Therefore, b2 - 4ac = 0
⟹ (6)2 - 4(p - 3) × 9 = 0
⟹ 36 - 36p + 108 = 0
⟹ 144 - 36p = 0
⟹ -36p = - 144
⟹ p = −144−36
⟹ p = 4
Therefore, the value of p = 4.
3. Without solving the equation 6x2 - 7x + 2 = 0, discuss the nature of its roots.
Solution:
Comparing 6x2 - 7x + 2 = 0 with ax2 + bx + c = 0 we have a = 6, b = -7, c = 2.
Therefore, discriminant = b2 – 4ac = (-7)2 - 4 ∙ 6 ∙ 2 = 49 - 48 = 1 > 0.
Therefore, the roots (solution) are real and unequal.
Note: Let a, b and c be rational numbers in the equation ax2 + bx + c = 0 and its discriminant b2 - 4ac > 0.
If b2 - 4ac is a perfect square of a rational number then b2−4ac−−−−−−−√ will be a rational number. So, the solutions x = −b±b2−4ac√2a will be rational numbers. But if b2 – 4ac is not a perfect square then b2−4ac−−−−−−−√ will be an irrational numberand as a result the solutions x = −b±b2−4ac√2a will be irrational numbers. In the above example we found that the discriminant b2 – 4ac = 1 > 0 and 1 is a perfect square (1)2. Also 6, -7 and 2 are rational numbers. So, the roots of 6x2 – 7x + 2 = 0 are rational and unequal numbers.