Problem 18 - Algebra 1

Question

Find the \(x\) -intercepts of the graph of the equation. $$y=2 x^{2}+4 x-30$$

Step-by-Step Solution

Answer

The x-intercepts of the graph of the equation are \(x = 3\) and \(x = -5\).

Step 1: Set the function equal to zero

Replace \(y\) with zero in the given function. This will give us the equation \[0 = 2x^{2} + 4x - 30\].

Step 2: Simplify the equation

Before proceeding to solve this equation, it can be simplified by dividing each term by the common factor of 2. This yields the simplified equation \[0 = x^{2} + 2x - 15.\]

Step 3: Factorize the equation

Factorize the equation \[0 = x^{2} + 2x - 15\] to give \[(x - 3)(x + 5) = 0.\]

Step 4: Solve for \(x\)

Finally, solve for \(x\) by setting each factor equal to zero. Thus, \(x - 3 = 0\), giving \(x = 3\), and \(x + 5 = 0\), giving \(x = -5\).

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). They are second-degree polynomials, meaning the highest power of \(x\) that appears is 2. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \(a\). In the case of our exercise \(y=2x^2+4x-30\), the coefficient \(2\) indicates that the parabola opens upwards.

When it comes to finding the x-intercepts of a quadratic function, we are essentially looking for the points where the parabola crosses the x-axis. These points correspond to the values of \(x\) where the function evaluates to zero. These intercepts are crucial for sketching the graph of the function and understanding its behavior.

Solving Quadratic Equations

Solving quadratic equations involves finding values of \(x\) that make the equation \(0 = ax^2 + bx + c\) true. To do this, we often start by setting the equation to zero, as we're interested in the points where the quadratic graph intersects the x-axis. These are the x-intercepts.

The standard methods for solving quadratics are factoring, completing the square, using the quadratic formula, and graphing. In our exercise, the equation has been factored to find the x-intercepts. However, if factoring is challenging or not possible, we would employ the quadratic formula \(x = {-b \pm \sqrt{{b^2-4ac}}}/{2a}\), where \(a\), \(b\), and \(c\) are from the quadratic equation's standard form.

Factorization of Polynomials

Factorization of polynomials is a method used to express a polynomial as a product of its factors. Factors are simpler polynomials (often binomials) whose product equals the original polynomial. For quadratic polynomials \(ax^2 + bx + c\), factorization involves finding two binomials \( (x - m)(x - n) \= 0 \) such that the product of \(m\) and \(n\) equals \(c\), and their sum equals \(b\).

Factoring is particularly useful for finding the roots or x-intercepts of quadratic functions. In the given exercise, the trinomial \(x^2 + 2x - 15\) was factored into \( (x - 3)(x + 5) \= 0 \), revealing the roots directly. It is one of the most powerful and commonly used techniques in algebra, especially when dealing with polynomials of lower degrees.

Quadratic Graphs

Quadratic graphs represent the set of points in the coordinate plane that satisfy the quadratic function's equation, forming a U-shaped curve called a parabola. The vertex of the parabola is the highest or lowest point, depending on whether it opens up or down. Features of the graph, such as the x-intercepts, the y-intercept, the vertex, and the axis of symmetry, help us better understand the function's behavior.

In our exercise, by finding the x-intercepts at \(x = 3\) and \(x = -5\), we located the points where the parabola crosses the x-axis. These intercepts, along with the vertex and y-intercept, are invaluable in sketching a complete and accurate graph of the quadratic function. Understanding quadratic graphs is crucial in numerous fields, including physics, engineering, and economics, as they often arise in natural phenomena and financial models.