Problem 38 - Precalculus Mathematics for Calculus

Question

Solve the equation both algebraically and graphically. $$x^{2}+3=2 x$$

Step-by-Step Solution

Answer

The equation has no real solutions, only complex roots: \(1 \pm i\sqrt{2}\).

Step 1: 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: \(x^2 - 2x + 3 = 0\).

Step 2: 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, \(a = 1\), \(b = -2\), \(c = 3\).

Step 3: 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.

Step 4: 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 \(x = 1 - i\sqrt{2}\).

Step 5: 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.

Key Concepts

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.

This formula is useful for solving any quadratic equation, especially when they do not factor neatly. You merely substitute the coefficients into the equation and solve step-by-step. In our example, we applied this formula to find the roots for the equation \(x^2 - 2x + 3 = 0\). When you solved it, you found that the result included complex numbers, which we will discuss more in the next section.

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 \(i = \sqrt{-1}\)

In our solved example, the complex roots found were \(x = 1 + i\sqrt{2}\) and \(x = 1 - i\sqrt{2}\). This indicates that there are no points at which the equation crosses the x-axis on a real-plane graph. Complex roots often appear in conjugate pairs, where one root is the mirror of the other in terms of sign of the imaginary part.

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.

In the provided problem, the discriminant \(b^2 - 4ac\) was \(-8\). This negative value confirmed the presence of complex roots. Calculating the discriminant provides a quick check to see if real roots exist and is helpful before delving deeper into solving the quadratic equation.

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.

Graphing can provide a quick confirmation of the nature of the roots and helps reinforce your algebraic findings. It is a practical method for visual learners and is often used to cross-verify solutions obtained algebraically.