Problem 63 - Algebra 1

Question

Find the x-intercepts of the graph of the equation. $$y=-x^{2}+4 x+1$$

Step-by-Step Solution

Answer

The x-intercepts of the function are \(x = 2 + \sqrt{5}\) and \(x = 2 - \sqrt{5}\).

Step 1: Set the Function to Zero

Set the function \(y = -x^2 + 4x + 1\) equal to zero to find the x-values when y = 0, i.e. \(0 = -x^2 + 4x + 1\).

Step 2: Rearrange the Equation

Rearrange the equation to the standard quadratic form, i.e. \(x^2 - 4x - 1 = 0.\)

Step 3: Solving the Quadratic Equation

Use the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for a quadratic equation in the form \(ax^2 + bx + c = 0\). In this case, \(a = 1\), \(b = -4\), and \(c = -1\). The solutions to the equation are \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4*1*(-1)}}{2*1} = 2 \pm \sqrt{5}\).

Step 4: Check the Solutions

Substitute \(x = 2 + \sqrt{5}\) and \(x = 2 - \sqrt{5}\) into the equation and confirm that they satisfy \(y = -x^2 + 4x + 1\).

Key Concepts

Quadratic Equation

A quadratic equation is an essential concept in algebra, commonly encountered in high school mathematics. It's a polynomial equation of degree two, which means the highest exponent of the variable is a square (i.e., 2). The general form of a quadratic equation is given as:\[ ax^2 + bx + c = 0 \]where:

• a, b, and c are constants with a ≠ 0.
• x is the variable or unknown.
• The term ax² is the quadratic term, bx is the linear term, and c is the constant term.

The solutions of a quadratic equation provide the x-values where the graph of the corresponding quadratic function intersects the x-axis, known as the x-intercepts or roots. It's crucial to transform any quadratic equation into this standard form before solving because it sets the stage for using the quadratic formula or other methods like factoring or completing the square.

Quadratic Formula

The quadratic formula is a powerful tool that provides the solutions to any quadratic equation written in standard form. It is particularly useful when the equation cannot be easily factored. The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's a breakdown of the components:

• a, b, and c are coefficients from the standard form of the quadratic equation \( ax^2 + bx + c = 0 \).
• The symbol \( \pm \) indicates there are generally two solutions, corresponding to the addition and subtraction of the square root term.
• The term under the square root, \( b^2 - 4ac \), is called the discriminant. It determines the nature of the roots:
  • If \( b^2 - 4ac > 0 \), there are two distinct real solutions.
  • If \( b^2 - 4ac = 0 \), there is one real solution, a repeated root.
  • If \( b^2 - 4ac < 0 \), the solutions are complex and not real numbers.

Understanding and using this formula will help you solve any quadratic equation efficiently, as exemplified in the problem solving process where the x-intercepts of the graph were found.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is essential for identifying and solving quadratic problems effectively. It is generally written as:\[ ax^2 + bx + c = 0 \]Here's why it's important:

• The arrangement helps in using various methods for solving the equation, like the quadratic formula, factoring, or completing the square.
• It makes it easy to identify coefficients a, b, and c, which are crucial inputs for the quadratic formula.
• Transforming any quadratic expression into this form often involves rearranging and simplifying the expression, as seen in the provided solution, where the equation was rearranged to \( x^2 - 4x - 1 = 0 \).

In any problem involving quadratic equations, recognizing and converting to the standard form is the first critical step towards finding the solution. This form provides a clear pathway to apply solving techniques effectively.