2.5 The Discriminant
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Learning Objectives
- Understand the concept of the discriminant in quadratic equations
- Calculate the discriminant of a quadratic equation
- Use the discriminant to determine the nature of roots
- Apply the discriminant to solve problems involving quadratic equations
- Understand the geometric interpretation of the discriminant
Key Concepts
Introduction to the Discriminant
The discriminant is a value calculated from the coefficients of a quadratic equation that provides information about the nature of its roots.
For a quadratic equation in the standard form ax² + bx + c = 0, the discriminant is given by the formula b² - 4ac.
The discriminant is a key component of the quadratic formula, which gives the solutions as x = (-b ± √(b² - 4ac))/(2a).
Calculating the Discriminant
To calculate the discriminant of a quadratic equation ax² + bx + c = 0, simply substitute the values of a, b, and c into the formula b² - 4ac.
The discriminant can be positive, zero, or negative, and each case tells us something different about the roots of the equation.
Example 1
Calculate the discriminant of the quadratic equation 2x² - 5x + 3 = 0.
Solution:
For the equation 2x² - 5x + 3 = 0, we have a = 2, b = -5, and c = 3. The discriminant is b² - 4ac = (-5)² - 4(2)(3) = 25 - 24 = 1.
Example 2
Calculate the discriminant of the quadratic equation 3x² + 2x + 4 = 0.
Solution:
For the equation 3x² + 2x + 4 = 0, we have a = 3, b = 2, and c = 4. The discriminant is b² - 4ac = 2² - 4(3)(4) = 4 - 48 = -44.
The Nature of Roots
The discriminant tells us about the nature of the roots of a quadratic equation:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one repeated real root (a double root).
- If b² - 4ac < 0, the equation has two complex conjugate roots (no real roots).
Example 1
Determine the nature of the roots of the equation x² - 6x + 9 = 0.
Solution:
For the equation x² - 6x + 9 = 0, we have a = 1, b = -6, and c = 9. The discriminant is b² - 4ac = (-6)² - 4(1)(9) = 36 - 36 = 0. Since the discriminant is zero, the equation has one repeated real root. We can find this root using the quadratic formula: x = (-b)/(2a) = 6/2 = 3. So the equation has a double root at x = 3.
Example 2
Determine the nature of the roots of the equation 2x² - 3x + 4 = 0.
Solution:
For the equation 2x² - 3x + 4 = 0, we have a = 2, b = -3, and c = 4. The discriminant is b² - 4ac = (-3)² - 4(2)(4) = 9 - 32 = -23. Since the discriminant is negative, the equation has two complex conjugate roots (no real roots). Using the quadratic formula, these roots are x = (3 ± √(-23))/(4), which can be written as x = 3/4 ± (√23)i/4.
Applications of the Discriminant
The discriminant has various applications in mathematics and problem-solving:
- Determining the number and nature of solutions to a quadratic equation without solving it
- Finding conditions for a quadratic equation to have specific types of roots
- Analyzing the behavior of quadratic functions and their graphs
Example 1
For what values of k will the equation x² + kx + 4 = 0 have equal roots?
Solution:
For the equation x² + kx + 4 = 0, we have a = 1, b = k, and c = 4. For the equation to have equal roots, the discriminant must be zero: b² - 4ac = 0 k² - 4(1)(4) = 0 k² - 16 = 0 k² = 16 k = ±4 So the equation will have equal roots when k = 4 or k = -4.
Example 2
For what values of p will the equation 2x² + px + 3 = 0 have no real roots?
Solution:
For the equation 2x² + px + 3 = 0, we have a = 2, b = p, and c = 3. For the equation to have no real roots, the discriminant must be negative: b² - 4ac < 0 p² - 4(2)(3) < 0 p² - 24 < 0 p² < 24 -√24 < p < √24 -4.9 < p < 4.9 So the equation will have no real roots when p is between -4.9 and 4.9.
Geometric Interpretation of the Discriminant
The discriminant has a geometric interpretation in terms of the graph of the quadratic function y = ax² + bx + c:
- If b² - 4ac > 0, the parabola crosses the x-axis at two distinct points (two real roots).
- If b² - 4ac = 0, the parabola touches the x-axis at exactly one point (one repeated root).
- If b² - 4ac < 0, the parabola does not intersect the x-axis (no real roots).
Example 1
Sketch the graph of y = x² - 4x + 3 and relate it to the discriminant.
Solution:
For the equation y = x² - 4x + 3, we have a = 1, b = -4, and c = 3. The discriminant is b² - 4ac = (-4)² - 4(1)(3) = 16 - 12 = 4 > 0. Since the discriminant is positive, the graph crosses the x-axis at two distinct points. We can find these points by solving x² - 4x + 3 = 0: (x - 1)(x - 3) = 0 x = 1 or x = 3 So the graph crosses the x-axis at x = 1 and x = 3. The vertex of the parabola is at x = -b/(2a) = 4/2 = 2, and y = f(2) = 2² - 4(2) + 3 = 4 - 8 + 3 = -1. So the vertex is at (2, -1). The parabola opens upward (since a > 0) and crosses the x-axis at two distinct points, which is consistent with the positive discriminant.
Example 2
Sketch the graph of y = x² - 6x + 9 and relate it to the discriminant.
Solution:
For the equation y = x² - 6x + 9, we have a = 1, b = -6, and c = 9. The discriminant is b² - 4ac = (-6)² - 4(1)(9) = 36 - 36 = 0. Since the discriminant is zero, the graph touches the x-axis at exactly one point. We can find this point by solving x² - 6x + 9 = 0: (x - 3)² = 0 x = 3 So the graph touches the x-axis at x = 3. The vertex of the parabola is at x = -b/(2a) = 6/2 = 3, and y = f(3) = 3² - 6(3) + 9 = 9 - 18 + 9 = 0. So the vertex is at (3, 0). The parabola opens upward (since a > 0) and touches the x-axis at exactly one point, which is consistent with the zero discriminant.
The Discriminant and Complex Roots
When the discriminant is negative, the quadratic equation has complex roots of the form p ± qi, where p and q are real numbers and i is the imaginary unit (i² = -1).
These complex roots always occur in conjugate pairs, meaning if p + qi is a root, then p - qi is also a root.
The real part of the complex roots is given by p = -b/(2a), and the imaginary part is given by q = √|b² - 4ac|/(2a).
Example 1
Find the complex roots of the equation x² + 2x + 5 = 0.
Solution:
For the equation x² + 2x + 5 = 0, we have a = 1, b = 2, and c = 5. The discriminant is b² - 4ac = 2² - 4(1)(5) = 4 - 20 = -16 < 0. Since the discriminant is negative, the equation has complex roots. Using the quadratic formula: x = (-b ± √(b² - 4ac))/(2a) = (-2 ± √(-16))/(2) = (-2 ± 4i)/2 = -1 ± 2i So the complex roots are x = -1 + 2i and x = -1 - 2i.
Example 2
Find the complex roots of the equation 2x² - 4x + 5 = 0.
Solution:
For the equation 2x² - 4x + 5 = 0, we have a = 2, b = -4, and c = 5. The discriminant is b² - 4ac = (-4)² - 4(2)(5) = 16 - 40 = -24 < 0. Since the discriminant is negative, the equation has complex roots. Using the quadratic formula: x = (-b ± √(b² - 4ac))/(2a) = (4 ± √(-24))/(4) = (4 ± 2√6i)/4 = 1 ± (√6/2)i So the complex roots are x = 1 + (√6/2)i and x = 1 - (√6/2)i.
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