To find the intercepts of the quadratic function [tex]\( y = x^2 + 4x - 5 \)[/tex] on the [tex]\( x \)[/tex]-axis, we need to determine the values of [tex]\( x \)[/tex] where [tex]\( y \)[/tex] equals zero, i.e., we need to solve the equation:
[tex]\[
x^2 + 4x - 5 = 0
\][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -5 \)[/tex]. The solutions to this quadratic equation can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, we identify and plug the values [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -5 \)[/tex] into the quadratic formula. Let's break this process down step-by-step:
1. Calculate the discriminant:
The discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot (-5) = 16 + 20 = 36
\][/tex]
Therefore, the discriminant [tex]\( \Delta \)[/tex] is [tex]\( 36 \)[/tex].
2. Calculate the two intercepts:
Using the quadratic formula, the two values of [tex]\( x \)[/tex] are:
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} \quad \text{and} \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a}
\][/tex]
Substituting [tex]\( b = 4 \)[/tex], [tex]\( a = 1 \)[/tex], and [tex]\( \sqrt{\Delta} = \sqrt{36} = 6 \)[/tex]:
[tex]\[
x_1 = \frac{-4 + 6}{2 \cdot 1} = \frac{2}{2} = 1
\][/tex]
[tex]\[
x_2 = \frac{-4 - 6}{2 \cdot 1} = \frac{-10}{2} = -5
\][/tex]
Therefore, the intercepts of the function [tex]\( y = x^2 + 4x - 5 \)[/tex] on the [tex]\( x \)[/tex]-axis are [tex]\( x = 1 \)[/tex] and [tex]\( x = -5 \)[/tex]. These are the points where the graph of the quadratic function crosses the [tex]\( x \)[/tex]-axis.
