Problem 59 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts of the graph of the function. $$y=x^{2}+5 x+6$$

Step-by-Step Solution

Answer

The x-intercepts of the function \(y = x^2 + 5x +6\) are \(x = -2\) and \(x = -3\)

Step 1: Set the function equal to zero

The x-intercepts are the x-values where the function equals zero. So, we need to solve the equation: \(x^{2} + 5x + 6 = 0 \)

Step 2: Factor the quadratic equation

As the equation is a quadratic equation, it can be factored into two binomial expressions. The factors of 6 which add up to 5 are 2 and 3. Therefore, the factored form of the equation is: \((x+2)(x+3)=0\)

Step 3: Solve for x

Set each factor equal to zero and solve for x: \(x + 2 = 0 \) gives \(x = -2\) and \(x + 3 = 0\) gives \(x = -3\) . These are the x-intercepts of the function.

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a foundational skill in algebra that allows us to find the values of the variable that make the equation true. Such equations are polynomial equations of the second degree, meaning they have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero.

The solutions to these equations, also known as the roots, can be found using several methods. One popular method is factoring, where we look for two binomial expressions that multiply to give the original quadratic equation. Other methods include completing the square, using the quadratic formula, or graphing the function to find the x-intercepts, where the graph crosses the x-axis. Each technique has its own advantages depending on the specific equation and context.

Factoring Quadratic Equations

Factoring quadratic equations involves writing the equation in a product of binomials. This technique is based on the principle that if the product of two numbers (or expressions) is zero, at least one of the numbers (or expressions) must be zero.

To factor a quadratic equation like \(x^2 + 5x + 6 = 0\), we search for two numbers that both add to the coefficient of \(x\) (which is 5 in this case) and multiply to the constant term (6 in this case). For our example, the numbers 2 and 3 meet these criteria since \(2 + 3 = 5\) and \(2 \times 3 = 6\). This gives us the factored form \((x + 2)(x + 3) = 0\).

The factoring technique is especially useful when the coefficients are integers and lend themselves to easy factorization. However, if the coefficients don't factor neatly, or if the quadratic is a prime polynomial, other methods such as the quadratic formula may be more appropriate.

Graphing Quadratic Functions

Graphing quadratic functions is an essential tool for visualizing the behavior and characteristics of these functions. A quadratic function graph is a parabola that can open upwards or downwards, depending on the sign of the leading coefficient, \(a\) in the standard form \(y = ax^2 + bx + c\).

The x-intercepts (also called zeros or roots) of the graph are the points where the parabola crosses the x-axis. To find these intercepts algebraically, we set the function equal to zero and solve for \(x\). Graphically, they're found by plotting the function on a coordinate grid and identifying the points where the curve touches the x-axis.

Knowing the x-intercepts is valuable not only for sketching the graph but also for understanding the solutions to the corresponding quadratic equation. For instance, if a parabola does not cross the x-axis, the quadratic equation has no real solutions.

Zero Product Property

The zero product property is a vital concept when working with quadratic equations. It states that if the product of two expressions is zero, at least one of the expressions must also be zero. In symbols, if \(a \times b = 0\), then either \(a = 0\) or \(b = 0\) (or both).

This property is used to find the solutions to a factored quadratic equation. After factoring, we set each binomial factor to zero and solve for the variable to find the x-intercepts of the function. For example, from the factored equation \((x + 2)(x + 3) = 0\), we use the zero product property to set \(x + 2 = 0\) and \(x + 3 = 0\), resulting in the solutions \(x = -2\) and \(x = -3\).

The zero product property is straightforward yet powerful, simplifying the process of finding solutions to quadratic equations and other polynomials when they can be factored into a product of simpler expressions.