Problem 178 - Elementary Algebra

Question

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=-x^{2}-14 x-49 $$

Step-by-Step Solution

Answer

The y-intercept is \( (0, -49) \) and the x-intercepts are \( (-7, 0) \).

Step 1: Identify the equation

The given equation is: \[ y = -x^{2} - 14x - 49 \]

Step 2: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \):\[y = -(0)^2 - 14(0) - 49 \]\[ y = -49 \]So, the y-intercept is \( (0, -49) \).

Step 3: Find the x-intercepts

To find the x-intercepts, set \( y = 0 \) in the equation and solve for \( x \):\[ 0 = -x^{2} - 14x - 49 \]Solve the quadratic equation \[ x^{2} + 14x + 49 = 0 \]Notice this is a perfect square trinomial:\[ (x + 7)^2 = 0 \]So, \( x + 7 = 0 \)Thus, \( x = -7 \).So, the x-intercepts are \( (-7, 0) \).

Step 4: Summarize the intercepts

The y-intercept is \( (0, -49) \) and the x-intercepts are \( (-7, 0) \).

Key Concepts

quadratic equations

A quadratic equation is a type of polynomial equation of the form:

\[ ax^2 + bx + c = 0 \]

Here,

• a, b, and c are constants where a eq 0
• x is the variable

Quadratic equations are also known as second-degree polynomials because the highest power of the variable is 2.

The graph of a quadratic equation is a parabola that opens upwards if a > 0 and downwards if a < 0.

The solutions to a quadratic equation, also known as the roots, can be found using various methods, such as factoring, completing the square, or the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

finding intercepts

Intercepts are points where a graph intersects the axes. They are key in graphing and understanding equations:

• Y-intercept: This is the point where the graph crosses the y-axis. To find the y-intercept, set x = 0 in the equation and solve for y. For the given equation y = -x^2 - 14x - 49, when x = 0, we get y = -49. So, the y-intercept is (0, -49).


• X-intercepts: These are points where the graph intersects the x-axis. To find the x-intercepts, set y = 0 in the equation and solve for x. For the equation 0 = -x^2 - 14x - 49, we solve the quadratic equation to get x = -7. So, the x-intercepts are (-7, 0).

Understanding intercepts helps in sketching the graph and analyzing the behavior of the function.

solving equations

Solving equations involves finding the values of the variable that make the equation true. For quadratic equations, there are several methods:

• Factoring: This method involves expressing the quadratic equation as a product of its factors. If a quadratic equation can be factored completely, then the solutions can be found by setting each factor to zero.


• Quadratic Formula: The quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] solves any quadratic equation. It is derived from the standard form of a quadratic equation.


• Completing the Square: This method involves rewriting the quadratic equation in such a way that one side becomes a perfect square trinomial. This makes it easier to solve for the roots.

In the given solution, factoring was used because it was evident that the equation \[ x^2 + 14x + 49 = 0 \] is a perfect square trinomial and could be simplified as \[ (x + 7)^2 = 0 \]. Solving that gives \[ x = -7 \].