Question

the equation whose roots are multiplied by 2 of those of 3x²+5x-2=0 is
1) 3x²+10x-8=0
2) x²-5x+6=0
3) x²+5x+6=0
4) x²-5x-6=0
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Answer 1

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,

$3(\frac{x}{2})^2+5\frac{x}{2}-2=0\\\;\\\frac{3x^2}{4}+\frac{5x}{2}-2=0\\\;\\\frac{3x^2+10x-8}{4}=0\\\;\\3x^2+10x-8=0$

Which is the required quadratic equation.

Answer 2

Given :-

• The equation whose roots are multiplied by 2 of those of 3x² + 5x - 2 = 0.

To Find :-

• What is the equation.

Solution :-

$\dashrightarrow$ $\sf 3{x}^{2} + 5x - 2 =\: 0$

Replace x by $\sf\bold{\dfrac{x}{2}}$ in the above equation to get the proper equation,

$\implies$ $\sf 3\bigg(\dfrac{x}{2}\bigg)^{2} + 5\bigg(\dfrac{x}{2}\bigg) - 2 =\: 0$

$\implies$ $\sf 3 \times \dfrac{{x}^{2}}{4} + \dfrac{5x}{2} - 2 =\: 0$

$\implies$ $\sf \dfrac{3{x}^{2}}{4} + \dfrac{5x}{2} - 2 =\: 0$

$\implies$ $\sf \dfrac{3{x}^{2} + 10x - 8}{4} =\: 0$

By doing cross multiplication we get,

$\implies$ $\sf\bold{\red{3{x}^{2} + 10x - 8 =\: 0}}$

$\therefore$ The equation is 3x² + 10x - 8 = 0 .

Hence, the correct options is option no (1) 3x² + 10x - 8 = 0.