Problem 65 - Precalculus : Building Concepts and Connections

Question

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$5 x^{2}-2 x+3=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = 0.2 \pm 0.74833i\) and these also correspond to the zeros of the quadratic function \(f(x) = 5x^{2}-2x+3\).

Step 1: Identify Coefficients

In the quadratic equation \(5x^{2}-2x+3=0\), the coefficients are \(a = 5\), \(b = -2\), and \(c = 3\).

Step 2: Apply the Quadratic Formula

Replace \(a\), \(b\), and \(c\) in the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with their corresponding values. Thus we get \(x = \frac{-(-2) \pm \sqrt{(-2)^{2}-4*5*3}}{2*5}\).

Step 3: Simplify the Result

Simplify the expression to find the roots of the equation. The expression can be simplified to \(x = \frac{2 \pm \sqrt{4-60}}{10}\), and further to \(x = \frac{2 \pm \sqrt{-56}}{10}\). Computing further, \(x = \frac{2 \pm 7.4833i}{10}\), and finally we get \(x = 0.2 \pm 0.74833i\).

Step 4: Relate to Zeros of the Function

Conclusion: The solutions to the quadratic equation are the same as the zeros of the quadratic function. Therefore, the zeros of the function \(f(x) = 5x^{2} -2x + 3\) are \(x = 0.2 \pm 0.74833i\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula is particularly useful when factoring is difficult or impossible. The quadratic formula is defined as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are the coefficients of the terms in the quadratic equation.
The term under the square root sign, \( b^2 - 4ac \), is called the discriminant. The discriminant tells us about the nature of the roots of the quadratic equation.

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is one double real root.
• If it's negative, the roots are complex or imaginary numbers.

This formula allows us to determine not only whether we get real or complex numbers but also exactly what those solutions are.

Complex Numbers

Complex numbers are numbers that comprise a real part and an imaginary part. They are very important when dealing with the square roots of negative numbers, which do not have real solutions. A complex number is usually written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) representing \( \sqrt{-1} \).
Imaginary numbers open up an entirely new way to understand mathematics, essential for solving many kinds of equations, including quadratic equations with negative discriminants.

• The imaginary unit \( i \) satisfies \( i^2 = -1 \).
• Complex conjugates are of the form \( a + bi \) and \( a - bi \), which are often observed as solutions in quadratic equations with complex roots.

In the context of our problem, because the discriminant \( -56 \) is negative, the solutions or zeros involve complex numbers. They are expressed as \( 0.2 \pm 0.74833i \).

Zeros of a Function

The zeros of a function are the points where the graph of the function intersects the x-axis. For a quadratic function \( f(x) = ax^2 + bx + c \), the zeros are the solutions to the equation \( ax^2 + bx + c = 0 \), meaning they make the entire function equal to zero.
Finding zeros is a crucial part of analyzing functions because they can also represent x-intercepts and roots.

• In our specific exercise, the quadratic function is \( f(x) = 5x^2 - 2x + 3 \).
• The solutions \( 0.2 \pm 0.74833i \) are the zeros of this function.
• Since these zeros are complex, the graph of this function does not actually cross the x-axis.

Complex zeros indicate that the function is entirely above or below the x-axis depending on the leading coefficient and vertex position. This result helps us graphically and conceptually understand the behavior of quadratic functions.