Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
AP EAMCET20226 Jul 2022Evening ShiftMathematicsQuadratic EquationActual

If a, b, c, d are real numbers such that a < b < c < d , then the roots of the equation (x-a)(x-c)+2(x-b)(x-d)=0 are

Options

  1. AReal and need not be distinct
  2. BReal and distinct
  3. CNon-real and distinct
  4. DNon-real and need not be distinct

Correct answer

B. Real and distinct

Step-by-step solution

aligned & Here, (x-a)(x-c)+2(x-b)(x-d) & =x^2-(a+c) x+a c+2 [x^2-(b+d) x+b d ] & =x^2-(a+c) x+a c+2 x^2-2(b+d) x+2 b d & =3 x^2-(a+2 b+c+2 d)+a c+2 b d & D=b^2-4 a c & =(a+2 b+c+2 d)^2-12(a c+2 b d) & =[(a+2 d)+(2 b+c)]^2-12(a c+2 b d) & =[(a+2 d)-(2 b+c)]^2+4(a+2 d)(2 b+c) . -12(a c+2 b d) & =[(a+2 d)-(2 b+c)]^2+4 a c+8 a b+16 b d & =[(a+2 d)-(2 b+c)]^2-8 a c-8 b d+8 a b+8 c d & =[(a+2 d)-(2 b+c)]^2+8(c-b)(d-a) aligned As, a 0 Roots are real and distinct.

Practice Quadratic Equation on Quantrex Academy →

More from Quadratic Equation

If is the common root of the quadratic equations x^2-5 x+4 a=0 , x^2-2 a x-8=0 , where a R , then the value of ^4- ^3+68 is 2025If , are the roots of x^2-5 x-6 =0 and , are the roots of x^2-5 x-6 =0 , then + + + = 2025If , , are the roots of the equation x ^3+ px ^2+ qx + r =0 , then ( + )( + )( + )= 2025Let f(x)=x^2+2 b x+2 c^2 and g(x)=-x^2-2 c x+b^2, x R . If b and c are nonzero real numbers such that f(x)> g(x) , then | c b | lies in the interval 2025If x ^2-4 ax +5+ a >0 for all x R whenever a ( , ) , then 4 + = 2025If , , are the roots of the equation x^3-12 x^2+k x-18=0 and one of them is thrice the sum of the other two roots, then ^2+ ^2+ ^2-k= 2025The polynomial equation of degree 5 whose roots are the roots of the equation x^5-3 x^4-x^3+11 x^2-12 x+4=0 each increased by 2, is 2025If the difference of the roots of the equation x^2-7 x+10=0 is same as the difference of the roots of the equation x ^2-17 x + k =0 , then a divisor of k is 2025 Full Quadratic Equation list All AP EAMCET PYQs