Problem 62 - 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. $$-x^{2}+x-5=0$$

Step-by-Step Solution

Answer

The solutions for the equation are \( x_1 = \frac{-1 + \sqrt{21}}{-2} \) and \( x_2 = \frac{-1 - \sqrt{21}}{-2} \). These are also the zeros of the function \( f(x) = -x^2 + x - 5 \).

Step 1: Identify the coefficients

Identify the coefficients in the equation. Here, \( a=-1 \), \( b=1 \), and \( c=-5 \).

Step 2: Utilize the quadratic formula

Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the corresponding values of a, b, c into this formula.

Step 3: Simplify the roots

Simplify the expression above to get the roots of the equation, these should be the solutions. You get: \( x_1 = \frac{-1 + \sqrt{21}}{-2} \) and \( x_2 = \frac{-1 - \sqrt{21}}{-2} \).

Step 4: Prove the solutions

Prove that these solutions are indeed the zeros of the function. Substitute \( x_1 \) and \( x_2 \) into the function \( f(x) = -x^2 + x - 5 \) and verify that the function evaluates to zero in both cases.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are expressions of the form \( ax^2+bx+c=0 \). To find the solutions—also known as roots—you can use the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here's why it works:

To derive this formula, one completes the square on the general quadratic equation to isolate \(x\). This process reveals that the quadratic formula provides the solutions by considering the coefficients of the quadratic equation \(a\), \(b\), and \(c\). Whenever you are confronted with a quadratic equation, just identify these coefficients and plug them into the quadratic formula.

Following this approach ensures you always have a methodical way of tackling quadratic equations, no matter how complex they appear.

Finding Zeros of Quadratic Functions

In the context of quadratic functions, the term 'zeros' refers to the \(x\)-values where the function equals zero. For the quadratic function \( f(x) = ax^2 + bx + c \), finding the zeros is synonymous with solving the corresponding quadratic equation \( ax^2 + bx + c = 0 \).

The zeros of a quadratic function have a practical interpretation: they represent the points at which the graph of the function crosses the \(x\)-axis. It's important to understand that finding the zeros of a function is an essential concept in many areas of mathematics and science, as it reveals the roots or solutions to the equation being studied.

Simplifying Quadratic Roots

Simplifying quadratic roots is the process of expressing the solutions of a quadratic equation in their simplest form. Once you've used the quadratic formula and obtained the roots in the form \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), simplification involves:

• Reducing any common factors in the fraction.
• Simplifying the square root, if possible.
• Expressing complex solutions in standard form if the discriminant (\(b^2 - 4ac\)) is negative.

For real numbers, the objective is to have the solutions in a form that is easy to interpret and use, without cumbersome expressions. This step by step approach makes it easier to understand the properties of the roots and how they relate to the graph of the quadratic function. For example, roots in their simplest form can quickly tell us how many real solutions exist and what their exact values are.