To solve the quadratic equation [tex]\(x^2 - 6x + 4 = 0\)[/tex] and express the solutions in the form [tex]\( a \pm \sqrt{b} \)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are prime numbers, follow these steps:
### Step 1: Identify the coefficients
The quadratic equation [tex]\(x^2 - 6x + 4 = 0\)[/tex] has the following coefficients:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -6\)[/tex]
- [tex]\(c = 4\)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values, we get:
[tex]\[
\Delta = (-6)^2 - 4(1)(4) = 36 - 16 = 20
\][/tex]
### Step 3: Find the solutions using the quadratic formula
The quadratic formula to solve [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the coefficients and the discriminant, we have:
[tex]\[
x = \frac{6 \pm \sqrt{20}}{2 \times 1} = \frac{6 \pm \sqrt{20}}{2}
\][/tex]
### Step 4: Simplify the expression
Simplify the solutions:
[tex]\[
x = \frac{6 \pm \sqrt{20}}{2} = \frac{6 \pm \sqrt{4 \times 5}}{2} = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5}
\][/tex]
### Step 5: Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]
The solutions can be written as [tex]\( a \pm \sqrt{b} \)[/tex], where:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 5\)[/tex]
Both [tex]\(3\)[/tex] and [tex]\(5\)[/tex] are prime numbers. Thus, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
\boxed{3 \text{ and } 5}
\][/tex]
The solutions to [tex]\(x^2 - 6x + 4 = 0\)[/tex] are [tex]\(3 + \sqrt{5}\)[/tex] and [tex]\(3 - \sqrt{5}\)[/tex].
