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Quadratic Equations

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) $3 x^{2} - 2 \sqrt{6} x + 2 = 0$
(vi) $(x - 2 a) (x - 2 b) = 4 a b$
(vii) $9 a^{2} b^{2} x^{2} - 24 a b c d x + 16 c^{2} d^{2} = 0, a \neq 0, b \neq 0$
(viii) $2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0$
(ix) $(b + c) x^{2} - (a + b + c) x + a = 0$

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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.

(vi) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(vii) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(viii) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(ix) 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. Determine the nature of the roots of the following quadratic equations:

(i)

a \neq 0, b \neq 0

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. In the following, determine whether the given quadratic equations have real roots and if so, find the roots:

(i) 16x² = 24x + 1
(ii) x² + x + 2 = 0
(iii)

\sqrt{3} x^{2} + 10 x - 8 \sqrt{3} = 0

(iv) 3x² − 2x + 2 = 0
(v)

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

3 a^{2} x^{2} + 8 a b x + 4 b^{2} = 0, a \neq 0

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

(viii) x² − 2x + 1 = 0
(ix)

2 x^{2} + 5 \sqrt{3} x + 6 = 0

\sqrt{2} x^{2} + 7 x + 5 \sqrt{2} = 0

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

(xii) 3x² − 5x + 2 = 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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