To find the zeroes of the quadratic function [tex]\( f(x) = x^2 - 12x + 31 \)[/tex] using the quadratic formula, follow these steps:
1. Identify coefficients: The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. Comparing this with [tex]\( f(x) = x^2 - 12x + 31 \)[/tex], we identify:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -12 \)[/tex]
- [tex]\( c = 31 \)[/tex]
2. Quadratic formula: The quadratic formula to solve for [tex]\( x \)[/tex] in [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant: The discriminant ([tex]\( \Delta \)[/tex]) is calculated using the expression [tex]\( b^2 - 4ac \)[/tex].
[tex]\[
\Delta = (-12)^2 - 4(1)(31) = 144 - 124 = 20
\][/tex]
4. Calculate the solutions: Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex] back into the quadratic formula.
[tex]\[
x = \frac{-(-12) \pm \sqrt{20}}{2(1)} = \frac{12 \pm \sqrt{20}}{2}
\][/tex]
5. Simplify the solutions: Simplify the expression [tex]\( \sqrt{20} \)[/tex]. Since [tex]\( 20 = 4 \times 5 \)[/tex], we can say [tex]\( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \)[/tex]. Therefore, the expression under the square root simplifies:
[tex]\[
x = \frac{12 \pm 2\sqrt{5}}{2}
\][/tex]
We can then factor out the common divisor in the numerator:
[tex]\[
x = \frac{12}{2} \pm \frac{2\sqrt{5}}{2} = 6 \pm \sqrt{5}
\][/tex]
So the zeroes of the function [tex]\( f(x) = x^2 - 12x + 31 \)[/tex] in simplest radical form are:
[tex]\[
x = 6 + \sqrt{5} \quad \text{and} \quad x = 6 - \sqrt{5}
\][/tex]
For numerical verification, these zeroes are approximately:
- [tex]\( x_1 \approx 8.2361 \)[/tex]
- [tex]\( x_2 \approx 3.7639 \)[/tex]
Thus, the solutions in simplest radical form are [tex]\( 6 + \sqrt{5} \)[/tex] and [tex]\( 6 - \sqrt{5} \)[/tex].
