Problem 64
Question
Find the real roots of the equation. \(x^{2}-2 x+5=0\).
Step-by-Step Solution
Verified Answer
The equation \(x^{2}-2 x+5=0\) doesn't have any real roots.
1Step 1: Understand the Quadratic Equation
A quadratic equation is in the form \(ax^2 + bx + c = 0\), where a, b, and c are constants. In this case, a=1, b=-2, and c=5.
2Step 2: Calculate the Discriminant
The discriminant is part of the quadratic formula and it is calculated as \(\Delta = b^2 - 4ac\). In this case, \(\Delta = (-2)^2 - 4*1*5 = 4 - 20 = -16.\)
3Step 3: Check the Discriminant
If the discriminant is less than zero, this means that the equation does not have real roots. Since the discriminant in this case is \(-16\), which is less than zero, the equation \(x^{2}-2 x+5=0\) has no real roots.
Key Concepts
Discriminant in Quadratic EquationsNature of Quadratic Equation RootsSolving Quadratic Equations
Discriminant in Quadratic Equations
Understanding the discriminant is key to analyzing quadratic equations. It comes from the quadratic formula, which is derived from the general form of a quadratic equation, \(ax^2 + bx + c = 0\). The discriminant, denoted as \(\Delta\), is determined by the formula \(\Delta = b^2 - 4ac\).
Its value is crucial because it indicates the nature of the roots the quadratic equation will have:
Let's use the discriminant to analyze the given exercise. With \(a = 1\), \(b = -2\), and \(c = 5\), the discriminant calculates to \(\Delta = (-2)^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16\). Since this value is negative, we can immediately conclude that \(x^2 - 2x + 5 = 0\) will not yield any real roots.
Its value is crucial because it indicates the nature of the roots the quadratic equation will have:
- If \(\Delta > 0\), the equation has two distinct real roots.
- If \(\Delta = 0\), the equation has exactly one real root, also known as a repeated or double root.
- If \(\Delta < 0\), the equation has no real roots; instead, it has two complex roots.
Let's use the discriminant to analyze the given exercise. With \(a = 1\), \(b = -2\), and \(c = 5\), the discriminant calculates to \(\Delta = (-2)^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16\). Since this value is negative, we can immediately conclude that \(x^2 - 2x + 5 = 0\) will not yield any real roots.
Nature of Quadratic Equation Roots
The discriminant not only tells us if real roots exist but also reveals the nature of these roots. The sign and value of the discriminant give important information:
In the context of our example \(x^2 - 2x + 5 = 0\), we've found the discriminant to be \(\Delta = -16\), pointing to complex roots. Specifically, these roots will be of the form \(p + qi\) and \(p - qi\), where \(p\) and \(q\) represent real numbers, and \(i\) is the imaginary unit. It's important to recognize that while these roots are not real numbers, they are equally valid within the realm of complex numbers and can be used in further mathematical computations.
- A positive discriminant suggests the roots are real and distinct.
- A zero discriminant implies a single repeated real root.
- A negative discriminant indicates that the roots are complex and occur in conjugate pairs.
In the context of our example \(x^2 - 2x + 5 = 0\), we've found the discriminant to be \(\Delta = -16\), pointing to complex roots. Specifically, these roots will be of the form \(p + qi\) and \(p - qi\), where \(p\) and \(q\) represent real numbers, and \(i\) is the imaginary unit. It's important to recognize that while these roots are not real numbers, they are equally valid within the realm of complex numbers and can be used in further mathematical computations.
Solving Quadratic Equations
When solving quadratic equations, one common method is to use the quadratic formula: \[x = \frac{-b \pm \sqrt{\Delta}}{2a}\]. This formula provides the roots of the equation \(ax^2 + bx + c = 0\) based on the values of \(a\), \(b\), and \(c\). The formula incorporates the discriminant \(\Delta\), and the term \(\pm\sqrt{\Delta}\) within the formula indicates whether there are two roots (when \(\Delta > 0\)), one root (when \(\Delta = 0\)), or complex roots (when \(\Delta < 0\)).
For our example, though we can't find real roots due to the negative discriminant, we can still spell out the process. To find complex roots, we would substitute our coefficients into the quadratic formula, accounting for the imaginary unit for the square root of a negative number: \[x = \frac{-(-2) \pm \sqrt{-16}}{2 \cdot 1} = \frac{2 \pm 4i}{2}\]. From this, we simplify to get the two complex roots: \(x = 1 + 2i\) and \(x = 1 - 2i\). This example demonstrates how the quadratic formula is a powerful tool for finding both real and complex roots.
For our example, though we can't find real roots due to the negative discriminant, we can still spell out the process. To find complex roots, we would substitute our coefficients into the quadratic formula, accounting for the imaginary unit for the square root of a negative number: \[x = \frac{-(-2) \pm \sqrt{-16}}{2 \cdot 1} = \frac{2 \pm 4i}{2}\]. From this, we simplify to get the two complex roots: \(x = 1 + 2i\) and \(x = 1 - 2i\). This example demonstrates how the quadratic formula is a powerful tool for finding both real and complex roots.
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Problem 64
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