Problem 64 - 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. $$-2 x^{2}+3 x-1=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(-2 x^{2} + 3 x - 1 = 0\) are \(x = \frac{1}{2}\) and \(x = 1\). These solutions also correspond to the zeros of the function \(f(x) = -2x^2 + 3x - 1\).

Step 1: Expressing the equation in general form

The quadratic equation can be represented in the general form \(ax^2 + bx + c = 0\) where in this case, \(a = -2\), \(b = 3\) and \(c = -1\).

Step 2: Using the Quadratic Formula

Once the equation is in standard form, we can use the quadratic formula to find its solutions. The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Where 'b' is the coefficient of 'x', 'a' is the coefficient of \(x^2\) and 'c' is the constant.

Step 3: Substitution and Calculation

Substituting our coefficients into the quadratic formula gives: \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4*-2*-1}}{2*-2}\). Simplifying the equation gives: \(x = \frac{3 \pm \sqrt{9 - 8}}{-4}\). This leads to two solutions when simplified: \(x = \frac{1}{2}\) and \(x = 1\)

Step 4: Relating the solutions to the zeros of the function

The solutions of a quadratic equation also represent the zeros of its corresponding function. In this case, the function \(f(x) = -2x^2 + 3x - 1\) has zeros at \(x = \frac{1}{2}\) and \(x = 1\). These are the x-values where the function equals zero.

Key Concepts

Zeros of a Function

In mathematics, zeros of a function are the points where the function's output value is zero. In simple terms, if you imagine a graph of a function, these zeros are where the graph crosses the x-axis. For quadratic functions like the one in this exercise, a zero is also referred to as a root or solution of the equation.

The function given here is \( f(x) = -2x^2 + 3x - 1 \). To find the zeros of this function, we solve the equation \( -2x^2 + 3x - 1 = 0 \).

By resolving this equation, we found two x-values where the function becomes zero, which are \( x = \frac{1}{2} \) and \( x = 1 \). These solutions indicate the points on a graph where the curve will cross the x-axis. Understanding zeros can help you visualize the behavior of the graph and how it interacts with the axes.

Quadratic Formula

The quadratic formula is a powerful tool to find the solutions of a quadratic equation, which is any equation of the form \( ax^2 + bx + c = 0 \). This formula is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It allows you to calculate the roots of any quadratic equation, provided you have the coefficients \( a \), \( b \), and \( c \).

Applying the quadratic formula to our equation \( -2x^2 + 3x - 1 = 0 \) involves these steps:

• Identify the coefficients: \( a = -2 \), \( b = 3 \), and \( c = -1 \).
• Substitute these values into the quadratic formula.
• Simplify the expression to find the roots.

After substitution and calculation, we derived that the roots are \( x = \frac{1}{2} \) and \( x = 1 \). The ‘±’ symbol in the quadratic formula means you get two possible solutions, revealing where the curve intersects the x-axis.

Solving Quadratic Equations

Solving quadratic equations means finding the x-values that satisfy the given quadratic equation. These are the solutions or "roots". Quadratic equations can usually be solved by several methods, including factoring, using the quadratic formula, or graphing.

In our exercise, we utilized the quadratic formula method, which is particularly useful when the quadratic equation is challenging to factor or involves complex numbers. Each quadratic equation can have two solutions, one solution, or sometimes no real solution. Real solutions are the points where the graph intersects the x-axis, like our solutions \( x = \frac{1}{2} \) and \( x = 1 \).

By solving a quadratic equation, we discover the essential values of 'x' that define the specific points on a graph where the curve meets or touches the x-axis. Knowing how to solve these equations efficiently empowers you to analyze and understand various quadratic relationships and their applications in real-world scenarios.