Advertisements
Advertisements
Questions
Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:
`3x^2 - 4sqrt3x + 4 = 0`
Determine the nature of the roots of the following quadratic equation:
`3x^2 - 4sqrt3x + 4 = 0`
Advertisements
Solution
`3x^2 - 4sqrt3x + 4 = 0`
Comparing it with ax2 + bx + c = 0, we get
a = 3, b = `-4sqrt3` and c = 4
Discriminant = b2 - 4ac
= `(-4sqrt3)^2 - 4(3)(4)`
= 48 - 48
= 0
As b2 - 4ac = 0,
Therefore, real roots exist for the given equation and they are equal to each other.
And the roots will be `(-b)/(2a) `
i.e, `(-(-4)sqrt3)/(2xx3) and (-(-4sqrt3))/(2xx3)`
= `(4sqrt3)/(2sqrt3 xx sqrt3) and (4sqrt3)/(2sqrt3 xxsqrt3)`
Therefore, the roots are `2/sqrt3 and 2/sqrt3.`
APPEARS IN
RELATED QUESTIONS
Find that non-zero value of k, for which the quadratic equation kx2 + 1 − 2(k − 1)x + x2 = 0 has equal roots. Hence find the roots of the equation.
Solve the quadratic equation 2x2 + ax − a2 = 0 for x.
If one root of the quadratic equation is `3 – 2sqrt5` , then write another root of the equation.
Without solving the following quadratic equation, find the value of ‘p’ for which the roots are equal.
px2 – 4x + 3 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
x2 - 2kx + 7k - 12 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
2x2 + kx + 3 = 0
Find the least positive value of k for which the equation x2 + kx + 4 = 0 has real roots.
Determine the nature of the roots of the following quadratic equation :
x2 +3x+1=0
Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x2 - 5x + 7 = 0
Find the value of k so that sum of the roots of the quadratic equation is equal to the product of the roots:
(k + 1)x2 + (2k + 1)x - 9 = 0, k + 1 ≠ 0.
Find the nature of the roots of the following quadratic equations: `x^2 - (1)/(2)x - (1)/(2)` = 0
Find the value(s) of m for which each of the following quadratic equation has real and equal roots: (3m + 1)x2 + 2(m + 1)x + m = 0
If `(1)/(2)` is a root of the equation `x^2 + kx - (5)/(4) = 0`, then the value of k is ______.
Discuss the nature of the roots of the following equation: 3x2 – 7x + 8 = 0
If –5 is a root of the quadratic equation 2x2 + px – 15 = 0, then:
If `1/2` is a root of the equation `"x"^2 + "kx" - (5/4)` = 0 then the value of k is:
If x2 (a2 + b2) + 2x (ac + bd) + c2 +d2 = 0 has no real roots, then:
Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals? Why?
Find whether the following equation have real roots. If real roots exist, find them.
5x2 – 2x – 10 = 0
