If the roots of the given quadratic equation are real and equal then find
the value of 'm'.
(m-12) x² + 2 (m-12) x + 2 = 0
Answers
EXPLANATION.
Roots of the quadratic equation are real and equal.
⇒ (m - 12)x² + 2(m - 12)x + 2 = 0.
As we know that,
⇒ D = Discriminant Or b² - 4ac.
For real and equal : D = 0.
⇒ [2(m - 12)²] - 4(m - 12)(2) = 0.
⇒ [4(m - 12)²] - 8(m - 12) = 0.
As we know that,
Formula of :
⇒ (x - y)² = x² + y² - 2xy.
Using this formula in equation, we get.
⇒ [4(m² + 144 - 24m)] - 8m + 96 = 0.
⇒ 4m² + 576 - 96m - 8m + 96 = 0.
⇒ 4m² - 104m + 672 = 0.
⇒ 4[m² - 26m + 168] = 0.
⇒ m² - 26m + 168 = 0.
Factorizes the equation into middle term splits, we get.
⇒ m² - 14m - 12m + 168 = 0.
⇒ m(m - 14) - 12(m - 14) = 0.
⇒ (m - 12)(m - 14) = 0.
⇒ m = 12 and m = 14.
MORE INFORMATION.
Nature of the roots of the quadratic expression.
(1) = Real and different, if b² - 4ac > 0.
(2) = Rational and different, if b² - 4ac is a perfect square.
(3) = Real and equal, if b² - 4ac = 0.
(4) = If D < 0 Roots are imaginary and unequal Or complex conjugate.
Answer:
Given :-
- (m - 12)x² + 2(m - 12)x + 2 = 0
To Find :-
- What is the value of m.
Solution :-
Given equation :
➲ (m - 12)x² + 2(m - 12)x + 2 = 0
where,
- a = m - 12
- b = 2(m - 12)
- c = 2
As we know that :
★ Discriminate (D) = b² - 4ac ★
↦ D = [2(m - 12)²] - 4 × m - 12 × 2
↦ D = 4(m² - 12)²] - 4 × 2(m - 12)
➦ D = 4(m² - 12)² - 8(m - 12)
Now, as we know that:
✯ (x - y)² = x² + y² - 2xy ✯
➺ According to the question by using the formula we get,
⇒ [ 4(m² + 144 - 24m) ] - 8(m - 12)
⇒ (4m² + 576 - 96m) - 8m + 96
⇒ 4m² + 576 - 96m - 8m + 96 = 0
⇒ 4m² - 96m - 8m + 576 + 96 = 0
⇒ 4m² - 104m + 672 = 0
➺ Now, by taking 4 as common we get,
⇒ 4(m² - 26m + 168) = 0
⇒ m² - 26m + 186 = 0 × 4
⇒ m² - 26m + 186 = 0
➺ Now, by doing middle term break we get,
⇒ m² - (14 + 12)m + 186 = 0
⇒ m² - 14m - 12m + 186 = 0
⇒ m(m - 14) - 12(m - 14) = 0
⇒ (m - 14)(m - 12) = 0
⇒ (m - 14) = 0
⇒ m - 14 = 0
➠ m = 14
➺ Either,
⇒ (m - 12) = 0
⇒ m - 12 = 0
➠ m = 12
∴ The value of m is 14 or 12.