To factor the quadratic function [tex]\( f(x) = x^2 - 17x + 70 \)[/tex] and convert it to intercept form, follow these steps:
1. Identify the coefficients in the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]. For the given function [tex]\( f(x) = x^2 - 17x + 70 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -17 \)[/tex]
- [tex]\( c = 70 \)[/tex]
2. Find the roots of the quadratic equation [tex]\( x^2 - 17x + 70 = 0 \)[/tex]. We use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-17)^2 - 4 \cdot 1 \cdot 70 = 289 - 280 = 9
\][/tex]
4. Solve for the roots:
[tex]\[
x = \frac{-(-17) \pm \sqrt{9}}{2 \cdot 1} = \frac{17 \pm 3}{2}
\][/tex]
This results in two solutions:
[tex]\[
x_1 = \frac{17 + 3}{2} = \frac{20}{2} = 10
\][/tex]
[tex]\[
x_2 = \frac{17 - 3}{2} = \frac{14}{2} = 7
\][/tex]
5. Write the intercept form using the roots [tex]\( x_1 = 10 \)[/tex] and [tex]\( x_2 = 7 \)[/tex]. The intercept form of a quadratic function is:
[tex]\[
f(x) = a(x - x_1)(x - x_2)
\][/tex]
Here, [tex]\( a = 1 \)[/tex], so:
[tex]\[
f(x) = (x - 10)(x - 7)
\][/tex]
Hence, the intercept form of the function is:
[tex]\[
f(x) = (x - 10)(x - 7)
\][/tex]
