**Nature of the Roots of a Quadratic Equation Worksheet :**

Worksheet given in this section will be much useful to the students who would like to practice problems on nature of the roots of a quadratic equation.

Before look at the worksheet, if you would like to know the stuff related to nature of the roots to a quadratic equation,

**Problem 1 :**

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

x^{2} + 5x + 6 = 0

**Problem 2 :**

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

2x^{2} - 3x - 1 = 0

**Problem 3 :**

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

x^{2} - 16x + 64 = 0

**Problem 4 :**

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

3x^{2} + 5x + 8 = 0

**Problem 5 :**

If the roots of the equation 2x^{2} + 8x - m³ = 0 are equal, then find the value of m.

**Problem 6 :**

If the roots of the equation x^{2} - (p + 4)x + 2p + 5 = 0 are equal, then find the value of p.

**Problem 7 :**

If the roots of the equation x^{2} + (2s - 1)x + s^{2 }= 0 are real, then find the value of a.

**Problem 8 :**

If the roots of the equation x^{2} - 16x + k = 0 are real and equal, then find the value of k.

**Problem 9 :**

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

x^{2} - 5x = 2(5x + 1)

**Problem 10 : **

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

1/(x+1) + 2/(x-4) = 2

**Problem 1 :**

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

x^{2} + 5x + 6 = 0

**Solution :**

The given quadratic equation is in the general form

ax^{2} + bx + c = 0

Then, we have a = 1, b = 5 and c = 6.

Find the value of the discriminant b^{2} - 4ac.

b^{2} - 4ac = 5^{2} - 4(1)(6)

b^{2} - 4ac = 25 - 24

b^{2} - 4ac = 1

Here, b^{2} - 4ac > 0 and also a perfect square.

So, the roots are real, unequal and rational.

**Problem 2 :**

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

2x^{2} - 3x - 1 = 0

**Solution :**

The given quadratic equation is in the general form

ax^{2} + bx + c = 0

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

Find the value of the discriminant b^{2} - 4ac.

b^{2} - 4ac = (-3)^{2} - 4(2)(-1)

b^{2} - 4ac = 9 + 8

b^{2} - 4ac = 17

Here, b^{2} - 4ac > 0, but not a perfect square.

So, the roots are real, unequal and irrational.

**Problem 3 :**

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

x^{2} - 16x + 64 = 0

**Solution :**

The given quadratic equation is in the general form

ax^{2} + bx + c = 0

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

Find the value of the discriminant b^{2} - 4ac.

b^{2} - 4ac = (-16)^{2} - 4(1)(64)

b^{2} - 4ac = 256 - 256

b^{2} - 4ac = 0

So, the roots are real, equal and rational.

**Problem 4 :**

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

3x^{2} + 5x + 8 = 0

**Solution :**

The given quadratic equation is in the general form

ax^{2} + bx + c = 0

Then, we have a = 3, b = 5 and c = 8.

Find the value of the discriminant b^{2} - 4ac.

b^{2} - 4ac = 5^{2} - 4(3)(8)

b^{2} - 4ac = 25 + 96

b^{2} - 4ac = 121

Here, b^{2} - 4ac > 0 and also a perfect square.

So, the roots are real, unequal and rational.

**Problem 5 :**

If the roots of the equation 2x^{2} + 8x - m³ = 0 are equal, then find the value of m.

**Solution :**

The given quadratic equation is in the general form

ax^{2} + bx + c = 0

Then, we have a = 2, b = 8 and c = -m^{3}.

Because the roots of the given equation are equal,

b^{2} - 4ac = 0

8^{2} - 4(2)(-m^{3}) = 0

64 + 8m^{3} = 0

Subtract 64 from each side.

8m^{3} = -64

Divide each side by 8.

m^{3} = -8

m^{3} = (-2)^{3}

m = -2.

So, the value of m is -2.

**Problem 6 :**

If the roots of the equation x^{2} - (p + 4)x + 2p + 5 = 0 are equal, then find the value of p.

**Solution :**

The given quadratic equation is in the general form

ax^{2} + bx + c = 0

Then, we have a = 1, b = -(p + 4) and c = (2p + 5).

Because the roots of the given equation are equal,

b^{2} - 4ac = 0

[-(p + 4)]^{2} - 4(1)(2p + 5) = 0

Simplify.

(p + 4)^{2} - 4(2p + 5) = 0

p^{2} + 8p + 16 -8p -20 = 0

p^{2} - 4 = 0

p^{2} - 2^{2} = 0

(p + 2)(p - 2) = 0

p + 2 = 0 or p - 2 = 0

p = -2 or p = 2

So, the value of p is ±2.

**Problem 7 :**

If the roots of the equation x^{2} + (2s - 1)x + s^{2 }= 0 are real, then find the value of a.

**Solution :**

The given quadratic equation is in the general form

ax^{2} + bx + c = 0

Then, we have a = 1, b = (2s - 1) and c = s^{2}.

Because the roots of the given equation are equal,

b^{2} - 4ac ≥ 0

(2s - 1)^{2} - 4(1)(s^{2}) ≥ 0

Simplify.

4s^{2} - 4s + 1 - 4s^{2} ≥ 0

-4s + 1 ≥ 0

-4s ≥ -1

Divide each side by -4.

s ≤ 1/4

So, the value of s is less than or equal to 1/4.

**Note : **

Whenever we multiply or divide both sides of an inequality by a negative number, we have to flip the inequality sign.

**Problem 8 :**

If the roots of the equation x^{2} - 16x + k = 0 are real and equal, then find the value of k.

**Solution :**

The given quadratic equation is in the general form

ax^{2} + bx + c = 0

Then, we have a = 1, b = -16 and c = k.

Because the roots of the given equation are equal,

b^{2} - 4ac = 0

(-16)^{2} - 4(1)(k) = 0

256 - 4k = 0

Subtract 256 from each side.

-4k = -256

Divide each side by -4.

k = 64

So, the value of k is 64.

**Problem 9 :**

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

x^{2} - 5x = 2(5x + 1)

**Solution :**

The given quadratic equation is not in the general form.

First, write the given quadratic equation in the general form.

x^{2} - 5x = 2(5x + 1)

x^{2} - 5x = 10x + 2

x^{2} - 15x - 2 = 0

Now, the quadratic equation is the general form

ax^{2} + bx + c = 0

Then, we have a = 1, b = -15 and c = -2.

Find the value of the discriminant b^{2} - 4ac.

b^{2} - 4ac = (-15)^{2} - 4(1)(-2)

b^{2} - 4ac = 225 + 8

b^{2} - 4ac = 233

Here, b^{2} - 4ac > 0, but not a perfect square.

So, the roots are real, unequal and irrational.

**Problem 10 : **

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

1/(x+1) + 2/(x-4) = 2

**Solution :**

The given quadratic equation is not in the general form.

First, write the given quadratic equation in the general form.

1/(x+1) + 2/(x-4) = 2

Add the two fractions on the right side of the equation using cross multiplication.

[(x-4) + 2(x+1)] / [(x+1)(x-4)] = 2

(x - 4 + 2x + 2) / (x^{2} - 3x - 4) = 2

(3x - 2) / (x^{2} - 3x - 4) = 2

Multiply each side by (x^{2} - 3x - 4).

3x - 2 = 2(x^{2} - 3x - 4)

3x - 2 = 2x^{2} - 6x - 8

2x^{2} - 9x - 6 = 0

Now, the quadratic equation is the general form

ax^{2} + bx + c = 0

Then, we have a = 2, b = -9 and c = -6.

Find the value of the discriminant b^{2} - 4ac.

b^{2} - 4ac = (-9)^{2} - 4(2)(-6)

b^{2} - 4ac = 81 + 48

b^{2} - 4ac = 129

Here, b^{2} - 4ac > 0, but not a perfect square.

So, the roots are real, unequal and irrational.

After having gone through the stuff given above, we hope that the students would have understood, "Nature of the Roots of a Quadratic Equation".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Time and work word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**

**Sum of all three four digit numbers formed using 0, 1, 2, 3**

**Sum of all three four digit numbers formed using 1, 2, 5, 6**

Widget is loading comments...