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Example 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 "b² - 4ac"
b² - 4ac = 5² - 4(1)(6)
b² - 4ac = 25 - 24
b² - 4ac = 1 (>0 and also a perfect square)
Hence, the roots are real, distinct and rational.
Let us look at the next example on "Nature of roots worksheet1"
Example 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 "b² - 4ac"
b² - 4ac = (-3)² - 4(2)(-1)
b² - 4ac = 9 + 8
b² - 4ac = 17 (>0 but not a perfect square)
Hence, the roots are real, distinct and irrational.
Let us look at the next example on "Nature of roots worksheet1"
Example 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 "b² - 4ac"
b² - 4ac = (-16)² - 4(1)(64)
b² - 4ac = 256 - 256
b² - 4ac = 0
Hence, the roots are real, equal and rational.
Let us look at the next example on "Nature of roots worksheet1"
Example 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 "b² - 4ac"
b² - 4ac = 5² - 4(3)(8)
b² - 4ac = 25- 96
b² - 4ac = -71 (negative)
Hence, the roots are imaginary.
Now we are going to look at some quiet different problems on "Nature of roots worksheet1".
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
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".
Let us look at the next example on "Nature of roots worksheet1"
Example 6 :
If the roots of the equation x² - (p+4)x + 2p + 5 = 0 are equal , then find the value of "p"
Solution :
If x² - (p+4)x + 2p + 5 = 0 is compared to the general form ax² + bx + c =0,
we get a = 1, b = - (p+4) and c = 2p+5
Since the roots are equal, we have
b² - 4ac = 0
[-(p+4)]² - 4(1)(2p+5) = 0
(p+4)² - 8p - 20 = 0
p² + 8p + 16 -8p -20 = 0
p² - 4 =0
p² = 4
p = ± 2
Hence, the value of "p" is " ±2 ".
Let us look at the next example on "Nature of roots worksheet1"
Example 7 :
If the roots of the equation x² + (2p-1)x + p² = 0 are real , then find the value of "p"
Solution :
If x² + (2p-1)x + p² = 0 is compared to the general form ax² + bx + c =0,
we get a = 1, b = 2p-1 and c = p²
Since the roots are real, we have
b² - 4ac ≥ 0
(2p-1)² - 4(1)(p²) ≥ 0
4p² - 4p +1 -4p² ≥ 0
- 4p +1 ≥ 0
1 ≥ 4p (or) 4p ≤ 1
p ≤ 1/4
Hence, the value of "p" is less than or equal to "1/4".
Let us look at the next example on "Nature of roots worksheet1"
Example 8 :
If the roots of the equation x² - 16x + k =0 are real and equal, then find the value of "k"
Solution :
If x² -16x + k = 0 is compared to the general form ax² + bx + c =0,
we get a = 1, b = -16 and c = k
Since the roots are real, we have
b² - 4ac = 0
(-16)² - 4(1)(k) = 0
64 - 4k = 0
64 = 4k
16 = k
(or) k = 4
Hence, the value of "k" is "4".
Let us look at the next example on "Nature of roots worksheet1"
Example 9 :
Examine the nature of the roots of the following quadratic equation.
x² - 5x = 2(3x+1)
Solution :
First, let us write the given equation in general form.
x² - 5x = 2(3x+1)
x² - 5x = 6x+2
x² - 11x -2 = 0
If x² -11x - 2 = 0 is compared to the general form ax² + bx + c =0,
we get a = 1, b = -11 and c = -2
Now, let us find the value of the discriminant "b² - 4ac"
b² - 4ac = (-11)² - 4(1)(-2)
b² - 4ac = 121 + 8
b² - 4ac = 129 (>0, but not a perfect square)
Hence, the roots are real, distinct and irrational
Let us look at the next example on "Nature of roots worksheet1"
Example 10 :
Examine the nature of the roots of the following quadratic equation. nature of the roots of quadratic equations worksheet pdf
Solution :
If 2x² - 9x -6 = 0 is compared to the general form ax² + bx + c =0,
we get a = 2, b = -9 and c = -6.
Now, let us find the value of the discriminant "b² - 4ac"
b² - 4ac = (-9)² - 4(2)(-6)
b² - 4ac = 81+ 48
b² - 4ac = 129 (>0, but not a perfect square)
Hence, the roots are real, distinct and irrational.
We hope that the student would have understood the problems and solutions given on "nature of roots worksheet1"
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Related Topics :
1. Nature of the roots of a quadratic equation (Detailed stuff)
2. Relationship between zeros and coefficients of a quadratic polynomial
3. Word problems on quadratic equation
4. Quadratic equation online solver
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