Advertisements
Advertisements
प्रश्न
Which of the following equations has no real roots?
विकल्प
`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`
Advertisements
उत्तर
`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.
संबंधित प्रश्न
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 value of p for which the quadratic equation (2p + 1)x2 − (7p + 2)x + (7p − 3) = 0 has equal roots. Also find these roots.
If one root of the quadratic equation is `3 – 2sqrt5` , then write another root of the equation.
Find the value of p, for which one root of the quadratic equation px2 – 14x + 8 = 0 is 6 times the other.
In the following determine the set of values of k for which the given quadratic equation has real roots:
2x2 + 3x + k = 0
If p, q are real and p ≠ q, then show that the roots of the equation (p − q) x2 + 5(p + q) x− 2(p − q) = 0 are real and unequal.
If 1 is a root of the quadratic equation 3x2 + ax – 2 = 0 and the quadratic equation a(x2 + 6x) – b = 0 has equal roots, find the value of b ?
Find the value of the discriminant in the following quadratic equation :
x2 +2x+4=0
Solve the following quadratic equation using formula method only
4 - 11 x = 3x2
Solve the following quadratic equation using formula method only
25x2 + 30x + 7 = 0
Solve the following quadratic equation using formula method only
3a2x2 +8abx + 4b2 = 0, a ≠ 0
Find the value of m for which the equation (m + 4)x2 + (m + 1)x + 1 = 0 has real and equal roots.
Find the value of k for which the given equation has real roots:
9x2 + 3kx + 4 = 0.
Without actually determining the roots comment upon the nature of the roots of each of the following equations:
3x2 + 2x - 1 = 0
If the ratio of the roots of the equation
lx2 + nx + n = 0 is p: q, Prove that
`sqrt(p/q) + sqrt(q/p) + sqrt(n/l) = 0.`
Find the discriminant of the following equations and hence find the nature of roots: 3x2 – 5x – 2 = 0
Discuss the nature of the roots of the following quadratic equations : `3x^2 - 2x + (1)/(3)` = 0
Find the values of p for which the equation 3x2 – px + 5 = 0 has real roots.
Choose the correct answer from the given four options :
If the equation 2x² – 5x + (k + 3) = 0 has equal roots then the value of k is
Discuss the nature of the roots of the following equation: `x^2 - (1)/(2)x - 4` = 0
If the roots of the given quadratic equation are real and equal, then find the value of ‘m’.
(m – 12)x2 + 2(m – 12)x + 2 = 0
If one root of the equation x2+ px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, the value of q is:
If p, q and r are rational numbers and p ≠ q ≠ r, then roots of the equation (p2 – q2)x2 – (q2 – r2)x + (r2 – p2) = 0 are:
State whether the following quadratic equation have two distinct real roots. Justify your answer.
x2 – 3x + 4 = 0
Find whether the following equation have real roots. If real roots exist, find them.
–2x2 + 3x + 2 = 0
If b and c are odd integers, then the equation x2 + bx + c = 0 has ______.
Assertion (A): If one root of the quadratic equation 4x2 – 10x + (k – 4) = 0 is reciprocal of the other, then value of k is 8.
Reason (R): Roots of the quadratic equation x2 – x + 1 = 0 are real.
Find the value of ‘c’ for which the quadratic equation
(c + 1) x2 - 6(c + 1) x + 3(c + 9) = 0; c ≠ - 1
has real and equal roots.
