Problem 61 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts of the graph of the function. $$y=x^{2}+10 x+16$$

Step-by-Step Solution

Answer

The x-intercepts of the function \(y=x^{2}+10x+16\) are x=-2 and x=-8

Step 1 - Set the Function Equal to Zero

To begin, the function \(y=x^{2}+10x+16\) is set equal to 0, which results in the equation \(x^{2}+10x+16=0\)

Step 2 - Apply the Quadratic Formula

The quadratic formula, which is \(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\), applies to any equation of the form ax^{2}+bx+c=0. In our case, a is 1, b is 10, and c is 16. Therefore, the formula becomes \(x=\frac{-10\pm \sqrt{(10)^{2}-4(1)(16)}}{2(1)}\) which simplifies to \(x=\frac{-10\pm \sqrt{100-64}}{2}\)

Step 3 - Simplify the Expression

The expression under the square root simplifies to 36, thus \(x=\frac{-10\pm \sqrt{36}}{2}\), this further simplifies to \(x=\frac{-10\pm 6}{2}\)

Step 4 - Solve for x

Now solve for the two possible solutions: \(x=\frac{-10+6}{2}=-2\) and \(x=\frac{-10-6}{2}=-8\)

Step 5 - Identify the x-intercepts

The x-intercepts of the graph of the function are the solutions to the equation, thus the x-intercepts are x=-2 and x=-8

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool when solving quadratic equations. A quadratic equation is any equation of the form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants.

When you can't easily factor the equation to find the roots, the quadratic formula can help you find the values of \( x \) that make the equation true. The formula itself is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, the expression under the square root, \( b^2 - 4ac \), is called the discriminant. The discriminant provides important information about the nature of the roots:

• If it is positive, you have two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If it is negative, there are no real solutions (instead, the solutions are complex numbers).

Always begin by identifying \( a \), \( b \), and \( c \) from your equation before using the quadratic formula.

X-Intercepts

X-intercepts are points where a graph crosses the x-axis. At these points, the value of \( y \) is zero.

To find the x-intercepts from a function like \( y = x^2 + 10x + 16 \), you set the function equal to zero and solve for \( x \). The solutions to this equation correspond to the x-intercepts of the graph.For example, in the quadratic equation \( x^2 + 10x + 16 = 0 \), once you solve it using the quadratic formula or another method like factoring, the values you obtain for \( x \) are your x-intercepts.
These are the points \( (-2, 0) \) and \( (-8, 0) \) on the graph. They indicate where the curve touches the x-axis.

Solving Equations

Solving quadratic equations involves finding the values of \( x \) that make the equation true. In general, there are several methods to solve quadratic equations, such as:

• Factoring the quadratic expression.
• Completing the square.
• Using the quadratic formula.

When you are working through a problem like \( x^2 + 10x + 16 = 0 \), you first need to decide which method is most suitable based on the equation's form and simplicity.

The quadratic formula is often the go-to method when the equation is hard to factor or when other methods seem cumbersome. After applying it and simplifying, you'll end up with the solutions, which are the roots of the equation. These solutions can then help you understand key characteristics of the graph, such as the location of the x-intercepts and the nature of the roots. Proper practice and familiarity with these methods enable you to solve quadratic equations efficiently.