**Using discriminant worksheet :**

Here we are going to see some practice questions using discriminant.

(1) Examine the nature of the roots of the following quadratic equation.

x² + 5x + 6 = 0

(2) Examine the nature of the roots of the following quadratic equation.

2x² - 3x + 1 = 0

(3) Examine the nature of the roots of the following quadratic equation.

x² - 16x + 64 = 0

(4) Examine the nature of the roots of the following quadratic equation.

3x² + 5x + 8 = 0

(5) If the roots of the equation 2x² + 8x - m³ = 0 are equal , then find the value of "m"

**Question 1 :**

Examine the nature of the roots of the following quadratic equation.

x² + 5x + 6 = 0

**Solution :**

If x² + 5x + 6 = 0 is compared to the general form ax² + bx + c = 0,

we get a = 1, b = 5 and c = 6.

Now, let us find the value of the discriminant

Δ = b² - 4ac

Δ = 5² - 4(1)(6)

Δ = 25 - 24

Δ = 1 > 0

Hence, the roots are real and unequal.

**Question 2 :**

Examine the nature of the roots of the following quadratic equation.

2x² - 3x + 1 = 0

**Solution :**

If 2x² - 3x + 1 = 0 is compared to the general form ax² + bx + c = 0,

we get a = 2, b = -3 and c = 1.

Now, let us find the value of the discriminant

Δ = b² - 4ac

Δ = (-3)² - 4(2)(-1)

Δ = 9 + 8

Δ = 17 > 0

Hence, the roots are real and unequal.

**Question 3 :**

Examine the nature of the roots of the following quadratic equation.

x² - 16x + 64 = 0

**Solution :**

If x² - 16x + 64 = 0 is compared to the general form ax² + bx + c = 0,

we get a = 1, b = -16 and c = 64.

Now, let us find the value of the discriminant

Δ = b² - 4ac

Δ = (-16)² - 4(1)(64)

Δ = 256 - 256

Δ = 0

Hence, the roots are real and equal.

**Question 4 :**

Examine the nature of the roots of the following quadratic equation.

3x² + 5x + 8 = 0

**Solution :**

If 3x² + 5x + 8 = 0 is compared to the general form ax² + bx + c = 0,

we get a = 3, b = 5 and c = 8.

Now, let us find the value of the discriminant

Δ = b² - 4ac

Δ = 5² - 4(3)(8)

Δ = 25- 96

Δ = -71 < 0

Hence, the roots are imaginary.

**Example 5 :**

If the roots of the equation 2x² + 8x - m³ = 0 are equal , then find the value of "m" ** **

**Solution :**

If 2x² + 8x - m³ = 0 is compared to the general form ax² + bx + c =0,

we get a = 2, b = 8 and c = -m³.

Since the roots are equal, we have Δ = 0

b² - 4ac = 0

8² - 4(2)(-m³) = 0

64 + 8m³ = 0

8m³ = -64

m³ = -8

m³ = (-2)³

m = - 2

Hence, the value of "m" is "-2".

After having gone through the stuff given above, we hope that the students would have understood "Using discriminant worksheet".

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