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Nature of Roots

Determine the...

Question

Determine the nature of the roots of the following quadratic equations:

(i) 2x² − 3x + 5 = 0
(ii) 2x² − 6x + 3 = 0
(iii) $\frac{3}{5} x^{2} - \frac{2}{3} x + 1 = 0$
(iv) $3 x^{2} - 4 \sqrt{3} x + 4 = 0$
(v) $3 x^{2} - 2 \sqrt{6} x + 2 = 0$

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Solution

(i) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(ii) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(iii) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(iv) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(v) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

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Similar questions

Q. Find the nature of the roots of the following quadratic equations:
(i)

2 x^{2} - 8 x + 5 = 0

3 x^{2} - 2 \sqrt{6} x + 2 = 0

5 x^{2} - 4 x + 1 = 0

5 x (x - 2) + 6 = 0

12 x^{2} - 4 \sqrt{15} x + 5 = 0

x^{2} - x + 2 = 0

Find the nature of the roots of the following quadratic equations:
(i) $2 x^{2} - 8 x + 5 = 0$

(ii) $3 x^{2} - 2 \sqrt{6} x + 2 = 0$
(iii) $5 x^{2} - 4 x + 1 = 0$

(iv) $5 x (x - 2) + 6 = 0$
(v) $12 x^{2} - 4 \sqrt{15} x + 5 = 0$

(vi) $x^{2} - x + 2 = 0$

Q. Find the roots of the quadratic equations by using the quadratic formula in the following equation:
(i)

2 x^{2} - 3 x - 5 = 0

5 x^{2} + 13 x + 8 = 0

- 3 x^{2} + 5 x + 12 = 0

- x^{2} + 7 x - 10 = 0

x^{2} + 2 \sqrt{2} x - 6 = 0

x^{2} - 3 \sqrt{5} x + 10 = 0

\frac{1}{2} x^{2} - \sqrt{11} x + 1 = 0

Find the nature of the roots of the following quadratic equations.

If the real roots exist, find them;

(I) 2x²−3x + 5 = 0

(II)

(III) 2x²− 6x + 3 = 0

Q. In the following, determine the set values of k for which the given quadratic equation has real roots:

(i)

2 x^{2} + 3 x + k = 0

2 x^{2} + k x + 3 = 0

2 x^{2} - 5 x - k = 0

k x^{2} + 6 x + 1 = 0

x^{2} - k x + 9 = 0

2 x^{2} + k x + 2 = 0

3 x^{2} + 2 x + k = 0

4 x^{2} - 3 k x + 1 = 0

2 x^{2} + k x - 4 = 0

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