Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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].
### 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].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.