If the quadratic equation 4 x2-2(a+c-1) x+a c-b=0(abc)

Question

If the quadratic equation 4 x2-2(a+c-1) x+a c-b=0(a>b>c) (A) Both roots are greater than a (B) Both roots are less than c (C) Both roots lie between c / 2 and a / 2 (D) Both one of the roots lies between c / 2 and a / 2.

Tardigrade Admin · Accepted Answer

Here, f(x)=(2 x-a)(2 x-c)+(2 x-b). So, f (( a /2))= a - b , f (( c /2))= c - b Now, f (( a /2)) f (( c /2))=( a - b )( c - b )<0( a > b > c ) Hence, exactly one of the roots lies between c / 2 and a / 2.

Q. If the quadratic equation $4 x^{2} - 2 (a + c - 1) x + a c - b = 0 (a > b > c)$

Both roots are greater than a

Both roots are less than c

Both roots lie between $c /2$ and $a /2$

Both one of the roots lies between $c /2$ and $a /2$ .

Here, $f (x) = (2 x - a) (2 x - c) + (2 x - b)$ . So,
$f (\frac{a}{2}) = a - b, f (\frac{c}{2}) = c - b$
Now,
$f (\frac{a}{2}) f (\frac{c}{2}) = (a - b) (c - b) < 0 (a > b > c)$
Hence, exactly one of the roots lies between $c /2$ and $a /2$ .

Q. If the quadratic equation 4x2−2(a+c−1)x+ac−b=0(a>b>c)

Both roots are greater than a

Both roots are less than c

Both roots lie between c/2 and a/2

Both one of the roots lies between c/2 and a/2.

Here, f(x)=(2x−a)(2x−c)+(2x−b). So, f(2a​)=a−b,f(2c​)=c−b Now, f(2a​)f(2c​)=(a−b)(c−b)<0(a>b>c) Hence, exactly one of the roots lies between c/2 and a/2.

Q. If the quadratic equation $4 x^{2} - 2 (a + c - 1) x + a c - b = 0 (a > b > c)$

Both roots lie between $c /2$ and $a /2$

Both one of the roots lies between $c /2$ and $a /2$ .

Here, $f (x) = (2 x - a) (2 x - c) + (2 x - b)$ . So,
$f (\frac{a}{2}) = a - b, f (\frac{c}{2}) = c - b$
Now,
$f (\frac{a}{2}) f (\frac{c}{2}) = (a - b) (c - b) < 0 (a > b > c)$
Hence, exactly one of the roots lies between $c /2$ and $a /2$ .