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Nature of Roots

In the follow...

Question

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) $2 x^{2} - 2 \sqrt{6} x + 3 = 0$
(vi) $3 a^{2} x^{2} + 8 a b x + 4 b^{2} = 0, a \neq 0$
(vii) $3 x^{2} + 2 \sqrt{5} x - 5 = 0$
(viii) x² − 2x + 1 = 0
(ix) $2 x^{2} + 5 \sqrt{3} x + 6 = 0$
(x) $\sqrt{2} x^{2} + 7 x + 5 \sqrt{2} = 0$
(xi) $2 x^{2} - 2 \sqrt{2} x + 1 = 0$
(xii) 3x² − 5x + 2 = 0

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Solution

In the following parts we have to find the real roots of the equations

(i) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Now we solve both cases for the two values of x. So, we have,

Also,

Therefore, the roots of the equation are and

(ii) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation does not satisfies this condition, hence it does not have real roots.

(iii) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Now we solve both cases for the two values of x. So, we have,

Also,

Therefore, the roots of the equation are and.

(iv) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation does not satisfies this condition, hence it does not have real roots.

(v) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real and equal roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Therefore, the roots of the equation are real and equal and its value is.

(vi) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Now we solve both cases for the two values of x. So, we have,

Also,

Therefore, the roots of the equation are and.

(vii) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Now we solve both cases for the two values of x. So, we have,

Also,

Therefore, the roots of the equation are and.

(viii) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real and equal roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Therefore, the roots of the equation are real and equal and its value is.

(ix) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Now we solve both cases for the two values of x. So, we have,

Also,

Therefore, the roots of the equation are and.

(x) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Now we solve both cases for the two values of x. So, we have,

Also,

Therefore, the roots of the equation are and

(xi) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real and equal roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Therefore, the roots of the equation are real and equal and its value is

(xii) We have been given,

Now we also know that for an equation, the discriminant is given by the following equation:

Now, according to the equation given to us, we have,, and.

Therefore, the discriminant is given as,

Since, in order for a quadratic equation to have real roots,.Here we find that the equation satisfies this condition, hence it has real roots.

Now, the roots of an equation is given by the following equation,

Therefore, the roots of the equation are given as follows,

Now we solve both cases for the two values of x. So, we have,

Also,

Therefore, the roots of the equation are and.

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Similar questions

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

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

Q. Write the discriminant of the following quadratic equations:

(i) 2x² − 5x + 3 = 0
(ii) x² + 2x + 4 = 0
(iii) (x − 1) (2x − 1) = 0
(iv) x² − 2x + k = 0, k ∈ R
(v)

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

(vi) x² − x + 1 = 0
(vii) (x + 5)² = 2(5x – 3)

Q. 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

3 x^{2} - 4 \sqrt{3} x + 4 = 0

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

(x - 2 a) (x - 2 b) = 4 a b

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

2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0

(b + c) x^{2} - (a + b + c) x + a = 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

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