Problem 5 - Introductory Algebra for College Students

Question

Find the \(x\) -intercepts for the parabola whose equation is given. If the \(x\) -intercepts are irrational numbers, round your answers to the nearest tenth. $$y=x^{2}-4 x+3$$

Step-by-Step Solution

Answer

The \(x\)-intercepts of the parabola given by the equation \(y=x^{2}-4x+3\) are \(x=1\) and \(x=3\).

Step 1: Identify the quadratic equation

The given equation is \(y = x^{2} - 4x +3\). To find the \(x\)-intercepts, set \(y=0\), which gives the equation to solve as \(x^2 - 4x +3 = 0\)

Step 2: Identify coefficients for the quadratic formula

The quadratic formula is \(-b \pm \sqrt{b^2 - 4ac} \over 2a\). The coefficients of the quadratic equation are a=1, b=-4, and c=3.

Step 3: Apply the quadratic formula

Substitute the values a=1, b=-4, and c=3 into the quadratic formula to solve for \(x\).

Step 4: Simplify to find the roots

Solving the quadratic formula gives the roots as \(x=1\) and \(x=3\). These are the \(x\)-intercepts of the given parabola.

Key Concepts

Understanding a Parabola

A parabola is a U-shaped curve that can open upwards or downwards. Its equation is typically in the form of a quadratic equation, like \( y = ax^2 + bx + c \). In the exercise we are discussing, the parabola is described by the equation \( y = x^2 - 4x + 3 \), which is a standard quadratic form where \( a = 1 \), \( b = -4 \), and \( c = 3 \).

Key features of a parabola include its vertex and axis of symmetry. The vertex is the peak or the lowest point of a parabola depending on its direction. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images. For the given equation, the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \). Performing the calculation gives \( x = 2 \). This lets you know the centerline of the parabola hasn't shifted horizontally.

• The parabola opens upwards because the coefficient \( a \) is positive.
• The vertex is at the point where \( x = 2 \) and can be found by substituting back into the equation.

Parabolas are essential in physics and engineering, often used to describe the paths of projectiles under gravity.

Finding X-intercepts

X-intercepts are the points where the parabola intersects the x-axis. These are the solutions to the equation when \( y = 0 \). For the equation \( y = x^2 - 4x + 3 \), finding the x-intercepts involves setting the equation to zero: \( x^2 - 4x + 3 = 0 \).

To identify these intercepts, we solve the equation for \( x \). This can be done by factoring, completing the square, or using the quadratic formula. Since our given equation is easily factorable, it factors into \( (x-1)(x-3) = 0 \). Setting each factor equal to zero gives us the intercepts \( x = 1 \) and \( x = 3 \).

• X-intercepts are also known as the roots or zeros of the quadratic equation.
• Each x-intercept represents the value of \( x \) where the parabola crosses the x-axis.

Understanding x-intercepts helps in graphing parabolas accurately and solving real-life problems involving projectile motion.

Using the Quadratic Formula

The quadratic formula is a powerful tool for finding the x-intercepts of any parabola, especially when it can't be factored easily. The formula is written as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our given equation \( x^2 - 4x + 3 = 0 \), the coefficients are \( a = 1 \), \( b = -4 \), and \( c = 3 \). By substituting these values into the quadratic formula, we calculate:

• First, the discriminant: \( b^2 - 4ac = (-4)^2 - 4 \times 1 \times 3 = 16 - 12 = 4 \).
• Then, solve for x: \( x = \frac{4 \pm \sqrt{4}}{2} = \frac{4 \pm 2}{2} \).

This results in the solutions \( x = 3 \) and \( x = 1 \). The quadratic formula not only finds the roots accurately but also helps to determine the nature of the roots (real or imaginary) through the discriminant. A positive discriminant, as seen here, indicates two real and distinct roots.

The quadratic formula is highly effective for solving any quadratic equation, reinforcing its critical role in algebra.