Absolutely! To find the x-intercepts of the given quadratic equations, we need to determine the roots of each equation. The x-intercepts, or roots, are the values of [tex]\( x \)[/tex] that make [tex]\( y \)[/tex] equal to zero. Here's a detailed step-by-step process for solving each quadratic equation:
### a. [tex]\( y = x^2 - 8x + 12 \)[/tex]
To find the x-intercepts:
1. Set [tex]\( y \)[/tex] to 0:
[tex]\[
0 = x^2 - 8x + 12
\][/tex]
2. Solve the quadratic equation:
[tex]\[
x^2 - 8x + 12 = 0
\][/tex]
3. Factorize the quadratic equation:
[tex]\[
(x - 2)(x - 6) = 0
\][/tex]
4. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\][/tex]
[tex]\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\][/tex]
So, the x-intercepts for [tex]\( y = x^2 - 8x + 12 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex].
### b. [tex]\( y = 3x^2 + 13x - 10 \)[/tex]
To find the x-intercepts:
1. Set [tex]\( y \)[/tex] to 0:
[tex]\[
0 = 3x^2 + 13x - 10
\][/tex]
2. Solve the quadratic equation:
[tex]\[
3x^2 + 13x - 10 = 0
\][/tex]
3. Factorize the quadratic equation. This may not always be straightforward, but let's say the factors are:
[tex]\[
(3x - 2)(x + 5) = 0
\][/tex]
4. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
3x - 2 = 0 \quad \Rightarrow \quad 3x = 2 \quad \Rightarrow \quad x = \frac{2}{3}
\][/tex]
[tex]\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\][/tex]
So, the x-intercepts for [tex]\( y = 3x^2 + 13x - 10 \)[/tex] are [tex]\( x = \frac{2}{3} \)[/tex] and [tex]\( x = -5 \)[/tex].
### c. [tex]\( y = x^2 - x - 20 \)[/tex]
To find the x-intercepts:
1. Set [tex]\( y \)[/tex] to 0:
[tex]\[
0 = x^2 - x - 20
\][/tex]
2. Solve the quadratic equation:
[tex]\[
x^2 - x - 20 = 0
\][/tex]
3. Factorize the quadratic equation:
[tex]\[
(x - 5)(x + 4) = 0
\][/tex]
4. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\][/tex]
[tex]\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\][/tex]
So, the x-intercepts for [tex]\( y = x^2 - x - 20 \)[/tex] are [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex].
### Summary
The x-intercepts for each quadratic equation are:
- For [tex]\( y = x^2 - 8x + 12 \)[/tex], the x-intercepts are [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex].
- For [tex]\( y = 3x^2 + 13x - 10 \)[/tex], the x-intercepts are [tex]\( x = \frac{2}{3} \)[/tex] and [tex]\( x = -5 \)[/tex].
- For [tex]\( y = x^2 - x - 20 \)[/tex], the x-intercepts are [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex].
These results indicate the values of [tex]\( x \)[/tex] where the graph of each quadratic function crosses the x-axis.
