if x²-8x+15=0 and (x-2)(x-p)=0 and one root is common, then p is
1) 1
2) 2
3) 3
4) 4
Answers
Answer:
Answer:
3x² + 10x - 8 = 0
Step-by-step explanation:
Let the roots of the given equation
3x² + 5x -2 = 0
be α and β
Sum of roots = -(Coefficient of x/Coefficient of x²)
α + β = -5/3
Product of roots = Constant term/Coefficient of x²
αβ = -2/3
Let the roots of required equation be λ and μ.
According to question
λ = 2α
and
μ = 2β
Thus the required quadratic equation will be,
x² - (λ + μ)x + λμ = 0
x² - (2α + 2β)x + 2α.2β = 0
x² - 2(α + β)x + 4αβ = 0
x² - 2(-5/3)x + 4(-2/3) = 0
x² + (10/3)x - (8/3) = 0
(3x² + 10x - 8)/3 = 0
3x² + 10x - 8 = 0
Which is the required quadratic equation.
Alternate Method:-
Given quadratic equation is
3x² + 5x - 2 = 0
Putting Replacing x with x/2,
\begin{gathered}3(\frac{x}{2})^2+5\frac{x}{2}-2=0\\\;\\\frac{3x^2}{4}+\frac{5x}{2}-2=0\\\;\\\frac{3x^2+10x-8}{4}=0\\\;\\3x^2+10x-8=0\end{gathered}
3(
2
x
)
2
+5
2
x
−2=0
4
3x
2
+
2
5x
−2=0
4
3x
2
+10x−8
=0
3x
2
+10x−8=0
Which is the required quadratic equation.
EXPLANATION.
If x² - 8x + 15 = 0. and (x - 2)(x - p) = 0 is one roots is common.
As we know that,
⇒ x² - 8x + 15 = 0.
Factorizes the equation into middle term splits, we get.
⇒ x² - 5x - 3x + 15 = 0.
⇒ x(x - 5) - 3(x - 5) = 0.
⇒ (x - 3)(x - 5) = 0.
⇒ x = 3 and x = 5.
Another equation,
⇒ (x - 2)(x - p) = 0.
Expand this equation, we get.
⇒ x² - px - 2x + 2p = 0.
⇒ x² - (p + 2)x + 2p = 0.
Put the value of x = 3 in equation, we get.
⇒ (3)² - (p + 2)(3) + 2p = 0.
⇒ 9 - [3p + 6] + 2p = 0.
⇒ 9 - 3p - 6 + 2p = 0.
⇒ 3 - p = 0.
⇒ p = 3.
Put the value of x = 5 in equation, we get.
⇒ (5)² - (p + 2)(5) + 2p = 0.
⇒ 25 - [5p + 10] + 2p = 0.
⇒ 25 - 5p - 10 + 2p = 0.
⇒ 15 - 3p = 0.
⇒ p = 5.
Values of p = 3 and p = 5.
Option [3] is correct answer.
MORE INFORMATION.
Conditions for common roots.
Let quadratic equations are a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0.
(1) = If only one root is common.
x = b₁c₂ - b₂c₁/a₁b₂ - a₂b₁.
y = a₂c₁ - a₁c₂/a₁b₂ - a₂b₁.
(2) = If both roots are common.
a₁/a₂ = b₁/b₂ = c₁/c₂.