Problem 65 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts of the graph of the function. $$y=2 x^{2}+4 x-6$$

Step-by-Step Solution

Answer

The x-intercepts of the function \(y = 2x^2 + 4x - 6\) are x = 1 and x = -3.

Step 1: Write the quadratic equation

The given function is \(y = 2x^2 + 4x - 6\). We will set y to 0 because we are looking for the x-intercepts, which are the values of x where y equals 0, thus get the equation \(2x^2 + 4x - 6 = 0\)

Step 2: Solve the quadratic equation

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For our equation, a = 2, b = 4, and c = -6. Substituting these values into the quadratic formula, we get \( x = \frac{-4 \pm \sqrt{4^2 - 4*2*(-6)}}{2*2} = \frac{-4 \pm \sqrt{16 + 48}}{4} = \frac{-4 \pm \sqrt{64}}{4} = \frac{-4 \pm 8}{4} \)

Step 3: Calculate the values of x

We have two solutions for x. The first solution is \( x = \frac{-4 + 8}{4} = 1 \) and the second solution is \( x = \frac{-4 - 8}{4} = -3 \). Therefore, the x-intercepts of the graph of the function \(y = 2x^2 + 4x - 6\) are x = 1 and x = -3.

Key Concepts

Quadratic Equation

A quadratic equation is an expression of degree 2, usually in the form \( ax^2 + bx + c = 0 \). In this formula, \( a \), \( b \), and \( c \) are constants, where \( a eq 0 \). This form is fundamental because it represents a parabola when graphed, which can open upwards or downwards depending on the sign of \( a \).
For example, the quadratic equation \( 2x^2 + 4x - 6 = 0 \) is given in this standard form with \( a = 2 \), \( b = 4 \), and \( c = -6 \).
Understanding how to write and interpret a quadratic equation is crucial because it allows one to find important characteristics of the parabola, like the x-intercepts, which are the points where the graph touches the x-axis.

Quadratic Formula

The quadratic formula is a tool for finding the x-intercepts of a quadratic equation. It is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula offers a way to find the solutions to any quadratic equation regardless of whether the roots are real or complex.
The formula relies on the piece under the square root called the discriminant, \( b^2 - 4ac \). This discriminant gives important information:

• If it is positive, the quadratic equation has two distinct real solutions.
• If it is zero, there is exactly one real solution (a repeated root).
• If it is negative, the equation has two complex solutions, and thus, no real x-intercepts.

By applying the quadratic formula to the equation \( 2x^2 + 4x - 6 = 0 \), we substitute \( a = 2 \), \( b = 4 \), and \( c = -6 \), to find the roots of the equation, giving us the x-intercepts \( x = 1 \) and \( x = -3 \).

Solving Equations

Solving equations, particularly quadratic ones, involves finding all values of the variable that make the equation true. For quadratic equations, this usually means identifying points where the equation equals zero.
There are several methods to solve quadratic equations:

• Factoring: This is useful when the quadratic can be easily expressed as a product of two binomials.
• Completing the Square: This method transforms the quadratic into a perfect square trinomial and is a step toward deriving the quadratic formula itself.
• Quadratic Formula: As discussed, this is a reliable method that can always be applied, especially when other methods are difficult or impossible.

In the case of the function \( y = 2x^2 + 4x - 6 \), using the quadratic formula provided a straightforward path to identify the x-intercepts by directly solving \( 2x^2 + 4x - 6 = 0 \). By calculating the values of \( x \), we determined where our parabola crosses the x-axis, which happened at \( x = 1 \) and \( x = -3 \). These intercepts are points of intersection that can be key for graph analysis and are critical for understanding the behavior and position of the graph on the coordinate plane.