Problem 60 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts of the graph of the function. $$y=x^{2}-11 x+24$$

Step-by-Step Solution

Answer

The x-intercepts of the graph are at \(x=8\) and \(x=3\).

Step 1: Set up the equation

Set the function equal to zero to find the x-intercepts. This gives the equation \(x^2-11x+24=0\)

Step 2: Factor the quadratic

Factor the quadratic equation. Two numbers that add up to -11 and multiply to 24 are -8 and -3. So the equation factors into \((x-8)(x-3)=0\).

Step 3: Solve for x

To solve for x, set each factor equal to zero and solve for x. This gives two x-intercepts : \(x-8=0=>x=8\) and \(x-3=0=>x=3\).

Key Concepts

Quadratic Functions

A quadratic function is a type of polynomial function that can be written in the standard form of y=ax^2+bx+c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, which can open either upwards or downwards depending on the sign of a. The highest or lowest point of the parabola is called the vertex, and the value of x at this point is the axis of symmetry of the graph.

Finding the x-intercepts of a quadratic function, often referred to as the roots or zeroes, is a fundamental aspect to understanding the function's graph. The x-intercepts are the points where the parabola crosses the x-axis, and these points play a crucial role in sketching the graph of the function. In the given exercise, the function y = x^2 - 11x + 24 is set to 0 to find these intercepts, since the y-value is 0 at the x-intercepts.

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations that can be re-written as the product of two binomials. For a quadratic equation ax^2+bx+c=0, factoring involves finding two numbers that both add together to give b and multiply together to give ac. When the quadratic is factorable, it can be expressed as (x-p)(x-q)=0, where p and q are the solutions to the equation. The two factors set equal to 0 yield the x-intercepts of the parabola.

In our example, we look for two numbers that add up to -11 and multiply to 24, resulting in these factors: -8 and -3. Consequently, the quadratic x^2-11x+24 is factored into (x-8)(x-3)=0. This step is crucial in making the equation easier to solve. Factoring is a skill that requires practice and can sometimes be approached through various methods if the quadratic does not factor neatly.

Solving Quadratic Equations

Solving quadratic equations can be accomplished by various methods, including factoring, using the quadratic formula, completing the square, or graphing. Factoring is often the simplest approach when it is possible. After factoring the equation into two binomials (x-p) and (x-q), you can use the Zero Product Property. It states that if the product of two factors equals zero, then at least one of the factors must be zero.

The final step is setting each binomial equal to zero, x-p=0 and x-q=0 and solving for x. These values of x are the solutions to the quadratic equation. In the problem (x-8)(x-3)=0, the solutions are x=8 and x=3, which are the x-intercepts of the parabola on the graph. Understanding the process of factoring and solving can dramatically improve students' abilities to solve and graph quadratic equations.