Problem 53 - Algebra 1: Concepts and Skills

Question

NUMBER OF X-INTERCEPTS Determine whether the graph of the function intersects the \(x\) -axis in zero, one, or two points. $$ y=x^{2}+x-10 $$

Step-by-Step Solution

Answer

The function \(y=x^{2}+x-10\) intersects the x-axis at 2 points: x = 3 and x = -4.

Step 1: Set the equation to zero

To find the x-intercepts, y-value must be zero, so form the equation \(x^{2}+x-10 = 0\).

Step 2: Factor the equation

Factor the equation into two binomials. If the equation is factorable, then solve for \(x\). The equation \(x^{2}+x-10 = 0\) can be factored into \((x-3)(x+4)=0\).

Step 3: Solve for x

Now we set each factor equal to zero and solve for x: \(x-3=0 \Rightarrow x=3\) and \(x+4=0 \Rightarrow x=-4\). Therefore, there are two x-intercepts and they are x = 3 and x = -4.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They are polynomial equations of degree two, which means the highest exponent of the variable is 2. A typical form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). One of the most notable features of quadratic equations is their U-shaped graph, known as a parabola.
To solve a quadratic equation, we need to find the values of \( x \) that make the equation true. These solutions correspond to the points where the parabola crosses the x-axis. We refer to these points as the x-intercepts.
Depending on the quadratic, it can intersect the x-axis in various ways:

• Zero points: The parabola does not cross the x-axis.
• One point: The vertex of the parabola touches the x-axis.
• Two points: The parabola crosses the x-axis at two distinct points.

This particular exercise deals with finding these x-intercepts by setting \( y = 0 \) to solve the equation \( x^2 + x - 10 = 0 \).

Factoring Quadratic Equations

Factoring is a key method used to solve quadratic equations when they are simple enough to be broken down. It involves expressing the quadratic equation as a product of two simpler expressions, often called binomials.
For our equation \( x^2 + x - 10 = 0 \), we want to write it in the form \( (x + m)(x + n) = 0 \). When expanded, this results in \( x^2 + (m+n)x + mn = 0 \). We find \( m \) and \( n \) such that their sum equals the coefficient of \( x \) (which is 1 in this case), and their product equals the constant term \(-10\).
By trial or knowledge, we determine \( m = -3 \) and \( n = 4 \), because \(-3 + 4 = 1\) and \(-3 \times 4 = -12\). Therefore, the quadratic equation can be factored as \((x - 3)(x + 4) = 0\).
Factoring is highly effective when applicable, but not all quadratic equations can be factored easily. In those cases, other methods such as completing the square or using the quadratic formula are necessary.

Solving Quadratic Equations

Once we have factored the quadratic equation, solving it involves finding the x-values that satisfy each factor set to zero. This step is straightforward after factoring.
For the factors \((x - 3)(x + 4) = 0\), we set each factor equal to zero:

• For the equation \(x - 3 = 0\), solving gives \(x = 3\).
• For the equation \(x + 4 = 0\), solving gives \(x = -4\).

These solutions \(x = 3\) and \(x = -4\) are the x-intercepts of the parabola described by the equation \(x^2 + x - 10\).
It's important to note that the number of solutions corresponds to the number of times the parabola intersects the x-axis. Since we have two solutions, the graph intersects at two points. Understanding this relationship between the equation and its graph helps in visualizing and verifying the solutions.