Problem 46
Question
Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=2 x-5$$
Step-by-Step Solution
Verified Answer
The equation has no real solutions; the discriminant is negative.
1Step 1: Rearrange the Equation
Start by expressing the given equation in standard quadratic form. The given equation is \(x^2 = 2x - 5\). Move all terms to one side to set the equation to zero:\[x^2 - 2x + 5 = 0\]
2Step 2: Identify Coefficients
Identify the coefficients of the quadratic equation \(ax^2 + bx + c = 0\). For our equation, \(a = 1\), \(b = -2\), and \(c = 5\).
3Step 3: Calculate the Discriminant
The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:\[D = b^2 - 4ac\]Substitute the identified coefficients into the formula:\[D = (-2)^2 - 4 \times 1 \times 5 = 4 - 20 = -16\]
4Step 4: Analyze the Discriminant
Since the discriminant \(D = -16\) is less than zero, the equation has no real solutions. This means the solutions are complex and cannot be represented on a real number graph.
5Step 5: Graphical Interpretation
Although the equation has no real solutions, you can still plot the parabola \(y = x^2 - 2x + 5\). Notice how the parabola does not intersect the x-axis, reinforcing that there are no real roots.
Key Concepts
Discriminant AnalysisComplex SolutionsGraphical Interpretation
Discriminant Analysis
Understanding the discriminant is pivotal in analyzing quadratic equations. The discriminant of a quadratic equation in the form \(ax^2 + bx + c = 0\) is calculated using the formula \(D = b^2 - 4ac\). This provides information on the nature of the equation's roots.
For the equation \(x^2 - 2x + 5 = 0\), the discriminant is computed as \((-2)^2 - 4 \times 1 \times 5\). It simplifies to \(4 - 20 = -16\).
When the discriminant is:
For the equation \(x^2 - 2x + 5 = 0\), the discriminant is computed as \((-2)^2 - 4 \times 1 \times 5\). It simplifies to \(4 - 20 = -16\).
When the discriminant is:
- Greater than zero, the equation has two distinct real roots.
- Equal to zero, the equation has exactly one real root (which we call a repeated or double root).
- Less than zero, as in this case, the equation has no real roots. Instead, it has two complex roots.
Complex Solutions
Quadratic equations may have complex solutions, especially when the discriminant is negative. Complex numbers are numbers that include the imaginary unit \(i\), where \(i^2 = -1\).
To find the complex solutions of our equation \(x^2 - 2x + 5 = 0\), we would use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Given our coefficients \(a = 1\), \(b = -2\), and \(c = 5\), we plug them into the formula: \[x = \frac{-(-2) \pm \sqrt{-16}}{2 \times 1}\] Simplifying yields: \[x = \frac{2 \pm 4i}{2}\] \[x = 1 \pm 2i\]
Thus, the solutions \(1 + 2i\) and \(1 - 2i\) illustrate the complex nature of the roots. In real-life terms, these indicate that the curve does not cross the x-axis.
To find the complex solutions of our equation \(x^2 - 2x + 5 = 0\), we would use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Given our coefficients \(a = 1\), \(b = -2\), and \(c = 5\), we plug them into the formula: \[x = \frac{-(-2) \pm \sqrt{-16}}{2 \times 1}\] Simplifying yields: \[x = \frac{2 \pm 4i}{2}\] \[x = 1 \pm 2i\]
Thus, the solutions \(1 + 2i\) and \(1 - 2i\) illustrate the complex nature of the roots. In real-life terms, these indicate that the curve does not cross the x-axis.
Graphical Interpretation
Understanding the graph of a quadratic function helps visualize the solutions. For the quadratic equation \(x^2 - 2x + 5 = 0\), its graph is a parabola.
A parabola is a symmetric curve that opens upwards if \(a > 0\) or downwards if \(a < 0\). In this case, since \(a = 1\), the parabola opens upwards.
The vertex of the parabola can be computed using the formula \((-b/2a, f(-b/2a))\). Here, \((-(-2)/2, f(1))\) provides the vertex as \((1, 4)\).
Because the discriminant is \(-16\), confirming no real solutions, the graph does not intersect the x-axis. Instead, it sits entirely above the x-axis, reinforcing the existence of complex roots. This graphical interpretation links the algebraic analysis (discriminant) to visual understanding, essential for mastering quadratic equations.
A parabola is a symmetric curve that opens upwards if \(a > 0\) or downwards if \(a < 0\). In this case, since \(a = 1\), the parabola opens upwards.
The vertex of the parabola can be computed using the formula \((-b/2a, f(-b/2a))\). Here, \((-(-2)/2, f(1))\) provides the vertex as \((1, 4)\).
Because the discriminant is \(-16\), confirming no real solutions, the graph does not intersect the x-axis. Instead, it sits entirely above the x-axis, reinforcing the existence of complex roots. This graphical interpretation links the algebraic analysis (discriminant) to visual understanding, essential for mastering quadratic equations.
Other exercises in this chapter
Problem 46
Use synthetic division to determine whether the given number is a zero of the polynomial. $$-1 ; \quad P(x)=x^{2}+4 x-5$$
View solution Problem 46
Add or subtract as indicated. Write each sum or difference in standard form. $$(4-i)+(2+5 i)$$
View solution Problem 46
Solve each equation and inequality. (a) \(4 x\left(x^{2}+1\right)\left(x^{2}+4\right)^{3}+6 x\left(x^{2}+1\right)^{2}\left(x^{2}+4\right)^{2}=0\) (b) \(4 x\left
View solution Problem 47
Use synthetic division to determine whether the given number is a zero of the polynomial. $$\text { 4; } P(x)=2 x^{3}-6 x^{2}-9 x+6$$
View solution