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Magazine Chapter 1: Problem 38
Solve the equation both algebraically and graphically. $$x^{2}+3=2 x$$
Short Answer
Expert verified
The equation has no real solutions, only complex roots: \(1 \pm i\sqrt{2}\).
Step by step solution
01
Rearrange the Equation
First, we need to rearrange the given equation into standard form. The equation given is \(x^{2} + 3 = 2x\). Subtract \(2x\) from both sides: \(x^2 - 2x + 3 = 0\).
02
Attempt to Solve Algebraically - Use the Quadratic Formula
The rearranged equation is \(x^2 - 2x + 3 = 0\). Since this quadratic does not factor neatly, we'll use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = -2\), \(c = 3\).
03
Calculate the Discriminant
Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots. Substituting the values gives \((-2)^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8\). The discriminant is negative, indicating there are no real roots, only complex roots.
04
Express the Roots Using Complex Numbers
With a negative discriminant, the roots are complex. Substitute into the quadratic formula: \(x = \frac{-(-2) \pm \sqrt{-8}}{2 \cdot 1} = \frac{2 \pm \sqrt{-8}}{2} = 1 \pm i\sqrt{2}\). So the roots are \(x = 1 + i\sqrt{2}\) and \(x = 1 - i\sqrt{2}\).
05
Verify Graphically - Plot the Equation
To solve graphically, plot \(y = x^2 - 2x + 3\) and \(y = 0\) which corresponds to the x-axis. Since the graph of \(y = x^2 - 2x + 3\) is a parabola opening upwards and its vertex is above the x-axis, it does not intersect the x-axis at any point. This confirms there are no real solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is one that can be written in the standard form \(ax^2 + bx + c = 0\). The quadratic formula used to find the solutions is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- The letters \(a\), \(b\), and \(c\) are coefficients from the equation.
- \( "+" \) and \( "-" \) before the square root indicate that there are generally two solutions.
Complex Roots
Complex roots occur in a quadratic equation when the discriminant, which is the part \(b^2 - 4ac\) of the quadratic formula, is negative. A negative discriminant implies that the square root of a negative number must be calculated, leading to complex or imaginary numbers. A complex number has two parts:
- The real part
- The imaginary part, denoted with the imaginary unit \(i\), where \(i = \sqrt{-1}\)
Discriminant
The discriminant is fundamental in determining the type of solutions you will get for a quadratic equation. It is calculated as:\[D = b^2 - 4ac\]
- If \(D > 0\), the equation has two distinct real roots.
- If \(D = 0\), there is exactly one real root, also known as a repeated or double root.
- If \(D < 0\), the roots are complex conjugates and no real intersections exist.
Graphical Methods
Graphical methods are a visual way to understand quadratic equations. When plotting a quadratic equation such as \(y = ax^2 + bx + c\), you generally get a parabolic graph:- If \(a > 0\), the parabola opens upwards.- If \(a < 0\), it opens downwards.In our example equation \(x^2 - 2x + 3 = 0\), plotting "\(y = x^2 - 2x + 3\)" involved drawing a parabola.
- The vertex of the parabola is above the x-axis, with no intersection, confirming no real solutions.
- The lack of any x-axis intersections visually indicates the roots are complex.
