Advertisements
Advertisements
Question
Which of the following equations has no real roots?
Options
`x^2 - 4x + 3sqrt(2) = 0`
`x^2 + 4x - 3sqrt(2) = 0`
`x^2 - 4x - 3sqrt(2) = 0`
`3x^2 + 4sqrt(3)x + 4 = 0`
Solution
`bb(x^2 - 4x + 3sqrt(2) = 0)`
Explanation:
(A) The given equation is `x^2 - 4x + 3sqrt(2)` = 0
On comparing with ax2 + bx + c = 0, we get
a = 1, b = – 4 and c = `3sqrt(2)`
The discriminant of `x^2 - 4x + 3sqrt(2)` = 0 is
D = b2 – 4ac
= `(-4)^2 - 4(1)(3sqrt(2))`
= `16 - 12sqrt(2)`
= 16 – 12 × (1.41)
= 16 – 16.92
= – 0.92
⇒ b2 – 4ac < 0
(B) The given equation is `x^2 + 4x - 3sqrt(2)` = 0
On comparing the equation with ax2 + bx + c = 0, we get
a = 1, b = 4 and c = `-3sqrt(2)`
Then, D = b2 – 4ac
= `(-4)^2 - 4(1)(-3sqrt(2))`
= `16 + 12sqrt(2) > 0`
Hence, the equation has real roots.
(C) Given equation is `x^2 - 4x - 3sqrt(2)` = 0
On comparing the equation with ax2 + bx + c = 0, we get
a = 1, b = – 4 and c = `-3sqrt(2)`
Then, D = b2 – 4ac
= `(-4)^2 - 4(1)(-3sqrt(2))`
= `16 + 12sqrt(2) > 0`
Hence, the equation has real roots.
(D) Given equation is `3x^2 + 4sqrt(3)x + 4` = 0
On comparing the equation with ax2 + bx + c = 0, we get
a = 3, b = `4sqrt(3)` and c = 4
Then, D = b2 – 4ac
= `(4sqrt(3))^2 - 4(3)(4)`
= 48 – 48
= 0
Hence, the equation has real roots.
Hence, `x^2 - 4x + 3sqrt(2)` = 0 has no real roots.
RELATED QUESTIONS
Solve for x: `1/(x+1)+2/(x+2)=4/(x+4), `x ≠ -1, -2, -3
Find the values of k for which the quadratic equation (k + 4) x2 + (k + 1) x + 1 = 0 has equal roots. Also find these roots.
Find that value of p for which the quadratic equation (p + 1)x2 − 6(p + 1)x + 3(p + 9) = 0, p ≠ − 1 has equal roots. Hence find the roots of the equation.
Find the values of k for which the roots are real and equal in each of the following equation:
4x2 - 3kx + 1 = 0
If the roots of the equation (b - c)x2 + (c - a)x + (a - b) = 0 are equal, then prove that 2b = a + c.
Determine the nature of the roots of the following quadratic equation :
2x2 + 5x - 6 = 0
Solve the following quadratic equation using formula method only
`5/4 "x"^2 - 2 sqrt 5 "x" + 4 = 0`
Solve the following quadratic equation using formula method only
5x2 - 19x + 17 = 0
Find the value of k for which the following equation has equal roots:
(k − 12)x2 + 2(k − 12)x + 2 = 0.
In each of the following determine the; value of k for which the given value is a solution of the equation:
kx2 + 2x - 3 = 0; x = 2
Determine whether the given values of x is the solution of the given quadratic equation below:
6x2 - x - 2 = 0; x = `(2)/(3), -1`.
Find the discriminant of the following equations and hence find the nature of roots: 3x2 – 5x – 2 = 0
Complete the following activity to find the value of discriminant for quadratic equation 4x2 – 5x + 3 = 0.
Activity: 4x2 – 5x + 3 = 0
a = 4 , b = ______ , c = 3
b2 – 4ac = (– 5)2 – (______) × 4 × 3
= ( ______ ) – 48
b2 – 4ac = ______
The sum of the roots of the quadratic equation 3x2 – 9x + 5 = 0 is:
If α + β = 4 and α3 + β3 = 44, then α, β are the roots of the equation:
State whether the following quadratic equation have two distinct real roots. Justify your answer.
x(1 – x) – 2 = 0
Solve for x: `5/2 x^2 + 2/5 = 1 - 2x`.
‘The sum of the ages of a boy and his sister (in years) is 25 and product of their ages is 150. Find their present ages.
