Problem 174 - Elementary Algebra

Question

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=x^{2}+10 x-11 $$

Step-by-Step Solution

Answer

The y-intercept is \(y = -11\). The x-intercepts are \(x = -11\) and \(x = 1\).

Step 1: Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). Substitute \(x = 0\) into the equation \(y = x^2 + 10x - 11\): \[y = (0)^2 + 10(0) - 11 = -11.\] Therefore, the y-intercept is \(y = -11\).

Step 2: Set up the equation to find the x-intercepts

To find the x-intercepts, set \(y = 0\) and solve for \(x\). The equation becomes: \[0 = x^2 + 10x - 11\]

Step 3: Factor the quadratic equation

Factor the quadratic equation \(x^2 + 10x - 11\) to find the values of \(x\). Find two numbers that multiply to -11 and add to 10, which are 11 and -1. Therefore, \[x^2 + 10x - 11 = (x + 11)(x - 1)\]

Step 4: Solve for the x-intercepts

Set each factor equal to zero and solve for \(x\): \[x + 11 = 0 \ => \ x = -11\] \[x - 1 = 0 \ => \ x = 1.\] Therefore, the x-intercepts are \(x = -11\) and \(x = 1\).

Key Concepts

Finding the y-intercepts

Finding the y-intercept of a quadratic equation is simple. The y-intercept is where the graph crosses the y-axis. To find it, set the value of x to 0. In other words, replace x with 0 in your equation and solve for y.

For the equation given:

• Start with:
  \[y = x^2 + 10x - 11\]
• Set \( x = 0\):
  \[y = (0)^2 + 10(0) - 11\]
• Simplify:
  \[y = -11\]

So, the y-intercept is at point \((0, -11)\), meaning when x is 0, y equals -11. This point shows where the graph meets the y-axis.

Finding the x-intercepts

The x-intercepts are the points where the graph crosses the x-axis. To find them, set y to 0 in the equation and solve for x. Here's how:

For the equation :

• Start with:
  \[y = x^2 + 10x - 11\]
• Set \( y = 0\):
  \[0 = x^2 + 10x - 11\]
• Now you have to solve this quadratic equation for x.

Factoring quadratics

To solve \(0 = x^2 + 10x - 11\), we factor the quadratic equation. Factoring is a way of rewriting the equation as a product of smaller expressions. Here, we look for two numbers that multiply to -11 and add to 10. Those numbers are 11 and -1.

So, rewrite the equation:

• Start with:
  \[x^2 + 10x - 11 = (x + 11)(x - 1)\]
• Next, set each factor equal to zero:
  • \(x + 11 = 0\)
    \(\Rightarrow x = -11\)
  • \(x - 1 = 0\)
    \(\Rightarrow x = 1\)
• So, the x-intercepts of the graph are \(x = -11\) and \(x = 1\). In point form, these intercepts are \((-11, 0)\) and \((1, 0)\).