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Byju's Answer

Standard IX

Mathematics

Factor of Polynomials

I: If x+2 i...

Question

I: If $x + 2$ is factor of $x^{2} + 2 a x + b$ and $x^{2} + 2 c x + d$ , then $\frac{b - d}{a - c} = 4$
II: The equation whose roots are the squares of the roots of $x^{2} + 4 x + 5 = 0$ is $x^{2} - 6 x + 25 = 0$
Which of the above statement(s) is/are true?

Only I

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Only II

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Both I and II

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Neither I nor II

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Solution

The correct option is B Both I and II
$x + 2$ is a factor $\Rightarrow x = - 2$ is a root of the given equations .
$4 - 4 a + b = 0; 4 - 4 c + d = 0$ $b - 4 a = d - 4 c$
$\Rightarrow \frac{b - d}{a - c} = 4$
Equation whose root are squares of $x^{2} + 4 x + 5$ is $(\sqrt{x})^{2} + 4 \sqrt{x} + 5 = 0$
$x + 5 = - 4 \sqrt{x}$
$(x + 5)^{2} = (4 \sqrt{x})^{2}$
$x^{2} + 25 x + 10 x = 16 x$
$x^{2} - 6 x + 25 = 0$

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Similar questions

Q. Statement I: lf

α, β

are the roots of

x^{2} - a x + b = 0

, then the equation whose roots are

\frac{α + β}{α}, \frac{α + β}{β}

b x^{2} - a^{2} x + a^{2} = 0

Statement II: lf

α, β

are the roots of

x^{2} - b x + c = 0

and

α + h, β + h

are the roots of

x^{2} + q x + r = 0

, then

h = b - q

.
Which of the above statement(s) is(are) true.

Q. Let

a, b, c, d

be distinct real numbers and

a

and

b

are the roots of quadratic equation

x^{2} - 2 c x - 5 d = 0

. If

c

and

d

are the roots of the quadratic equation

x^{2} - 2 a x - 5 b = 0

.then find the numerical values of

a + b + c + d

Q. (i) If

α, β

are the roots of

x^{2} + K x + 2 = 0

, and

α - β = 1

then

K = \pm 3

(ii) The difference of the roots of

x^{2} - b x + c = 0

is same as the difference of the roots of

x^{2} - c x + b = 0

. If

b \neq c

, then

b + c = - 3

.
Which of the above statements is correct.

The abscissa and ordinate of the points A and B are the roots of the equation $x^{2} + 2 a x + b = 0$ and $x^{2} + 2 c x + d = 0$ respectively, then equation of circle with AB as diameter is

Q. (i) lf

α, β

are the roots of

a x^{2} + b x + c = 0

. lf

K

is a real number

[K \neq 0, 1]

then the equation whose roots are

K α, K β

a x^{2} +

Kbx

+ K^{2} c = 0

.
(ii) 8,2 are roots of

x^{2} + a x + β = 0

and 3, 3 are the roots of

x^{2} + α x + b = 0

then the roots of

x^{2} + a x + b = 0

are 9, 1.
Which of the above statement is correct

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I: If x+2 is factor of x2+2ax+b and x2+2cx+d, then b−da−c=4II: The equation whose roots are the squares of the roots of x2+4x+5=0 is x2−6x+25=0 Which of the above statement(s) is/are true?

The correct option is B Both I and IIx+2 is a factor ⇒x=−2 is a root of the given equations .4−4a+b=0;4−4c+d=0b−4a=d−4c⇒b−da−c=4Equation whose root are squares of x2+4x+5 is(√x)2+4√x+5=0x+5=−4√x(x+5)2=(4√x)2x2+25x+10x=16xx2−6x+25=0

I: If $x + 2$ is factor of $x^{2} + 2 a x + b$ and $x^{2} + 2 c x + d$ , then $\frac{b - d}{a - c} = 4$
II: The equation whose roots are the squares of the roots of $x^{2} + 4 x + 5 = 0$ is $x^{2} - 6 x + 25 = 0$
Which of the above statement(s) is/are true?