Answer :
Certainly! Let's tackle this problem step-by-step.
### Part 1: Solve the quadratic equation [tex]\(x^2 - 8x + 3 = 0\)[/tex] by completing the square
1. Rewrite the equation:
[tex]\[ x^2 - 8x + 3 = 0 \][/tex]
Move the constant term to the right side:
[tex]\[ x^2 - 8x = -3 \][/tex]
2. Complete the square:
- Take half of the coefficient of [tex]\(x\)[/tex], which is [tex]\(-8\)[/tex]. Half of [tex]\(-8\)[/tex] is [tex]\(-4\)[/tex].
- Square [tex]\(-4\)[/tex] to get [tex]\(16\)[/tex].
Add [tex]\(16\)[/tex] to both sides:
[tex]\[ x^2 - 8x + 16 = -3 + 16 \][/tex]
Simplifying the right side:
[tex]\[ x^2 - 8x + 16 = 13 \][/tex]
3. Rewrite the left side as a squared term:
[tex]\[ (x - 4)^2 = 13 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Take the square root of both sides:
[tex]\[ x - 4 = \pm \sqrt{13} \][/tex]
So, we have two solutions:
[tex]\[ x = 4 + \sqrt{13} \quad \text{and} \quad x = 4 - \sqrt{13} \][/tex]
5. Convert solutions to decimal form:
- For [tex]\( x = 4 + \sqrt{13} \)[/tex]:
[tex]\[ x \approx 7.60555127546399 \][/tex]
- For [tex]\( x = 4 - \sqrt{13} \)[/tex]:
[tex]\[ x \approx 0.3944487245360107 \][/tex]
6. Round to one decimal place:
- [tex]\( x = 4 + \sqrt{13} \approx 7.6 \)[/tex]
- [tex]\( x = 4 - \sqrt{13} \approx 0.4 \)[/tex]
So, the solutions to the quadratic equation [tex]\(x^2 - 8x + 3 = 0\)[/tex] rounded to one decimal place are [tex]\( x \approx 7.6 \)[/tex] and [tex]\( x \approx 0.4 \)[/tex].
### Part 2: Simplify the expression [tex]\(\frac{ab}{3a-65} \times \frac{4a-8b}{a^2b}\)[/tex]
1. Write down the given expression:
[tex]\[ \frac{ab}{3a-65} \times \frac{4a-8b}{a^2b} \][/tex]
2. Combine the fractions:
Multiply the numerators and multiply the denominators:
[tex]\[ \frac{ab \cdot (4a - 8b)}{(3a - 65) \cdot a^2b} \][/tex]
3. Simplify the numerator and denominator:
The numerator is:
[tex]\[ ab \cdot (4a - 8b) = 4a^2b - 8ab^2 \][/tex]
The denominator is:
[tex]\[ a^2b \cdot (3a - 65) = a^3b (3a - 65) \][/tex]
4. Divide the terms:
Rewrite the expression by dividing each term in the numerator by the corresponding term in the denominator:
[tex]\[ \frac{4a^2b - 8ab^2}{a^3b(3a - 65)} = \frac{4a^2b}{a^3b(3a - 65)} - \frac{8ab^2}{a^3b(3a - 65)} \][/tex]
Simplifying each term separately:
[tex]\[ \frac{4a^2b}{a^3b(3a - 65)} = \frac{4a^2b}{a^3b * (3a - 65)} = \frac{4a^2}{a^3 (3a - 65)} = \frac{4}{a (3a - 65)} \][/tex]
[tex]\[ \frac{8ab^2}{a^3b(3a - 65)} = \frac{8ab^2}{a^3b (3a - 65)} = \frac{8b}{a^2 (3a - 65)} \][/tex]
5. Further simplification:
Simplifying the combined expression:
[tex]\[ \frac{4a^2b - 8ab^2}{a^3b (3a - 65)} = \frac{4a (a - 2b)}{a^2 (3a - 65)} = \frac{4(a - 2b)}{a (3a - 65)} \][/tex]
So, the simplified form of the expression is:
[tex]\[ \frac{4(a - 2b)}{a(3a - 65)} \][/tex]
Therefore, the final answers are:
1. The solutions to [tex]\( x^2 - 8x + 3 = 0 \)[/tex] are [tex]\( x \approx 7.6 \)[/tex] and [tex]\( x \approx 0.4 \)[/tex].
2. The simplified form of [tex]\(\frac{ab}{3a-65} \times \frac{4a-8b}{a^2b}\)[/tex] is [tex]\(\frac{4(a - 2b)}{a(3a - 65)}\)[/tex].
### Part 1: Solve the quadratic equation [tex]\(x^2 - 8x + 3 = 0\)[/tex] by completing the square
1. Rewrite the equation:
[tex]\[ x^2 - 8x + 3 = 0 \][/tex]
Move the constant term to the right side:
[tex]\[ x^2 - 8x = -3 \][/tex]
2. Complete the square:
- Take half of the coefficient of [tex]\(x\)[/tex], which is [tex]\(-8\)[/tex]. Half of [tex]\(-8\)[/tex] is [tex]\(-4\)[/tex].
- Square [tex]\(-4\)[/tex] to get [tex]\(16\)[/tex].
Add [tex]\(16\)[/tex] to both sides:
[tex]\[ x^2 - 8x + 16 = -3 + 16 \][/tex]
Simplifying the right side:
[tex]\[ x^2 - 8x + 16 = 13 \][/tex]
3. Rewrite the left side as a squared term:
[tex]\[ (x - 4)^2 = 13 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Take the square root of both sides:
[tex]\[ x - 4 = \pm \sqrt{13} \][/tex]
So, we have two solutions:
[tex]\[ x = 4 + \sqrt{13} \quad \text{and} \quad x = 4 - \sqrt{13} \][/tex]
5. Convert solutions to decimal form:
- For [tex]\( x = 4 + \sqrt{13} \)[/tex]:
[tex]\[ x \approx 7.60555127546399 \][/tex]
- For [tex]\( x = 4 - \sqrt{13} \)[/tex]:
[tex]\[ x \approx 0.3944487245360107 \][/tex]
6. Round to one decimal place:
- [tex]\( x = 4 + \sqrt{13} \approx 7.6 \)[/tex]
- [tex]\( x = 4 - \sqrt{13} \approx 0.4 \)[/tex]
So, the solutions to the quadratic equation [tex]\(x^2 - 8x + 3 = 0\)[/tex] rounded to one decimal place are [tex]\( x \approx 7.6 \)[/tex] and [tex]\( x \approx 0.4 \)[/tex].
### Part 2: Simplify the expression [tex]\(\frac{ab}{3a-65} \times \frac{4a-8b}{a^2b}\)[/tex]
1. Write down the given expression:
[tex]\[ \frac{ab}{3a-65} \times \frac{4a-8b}{a^2b} \][/tex]
2. Combine the fractions:
Multiply the numerators and multiply the denominators:
[tex]\[ \frac{ab \cdot (4a - 8b)}{(3a - 65) \cdot a^2b} \][/tex]
3. Simplify the numerator and denominator:
The numerator is:
[tex]\[ ab \cdot (4a - 8b) = 4a^2b - 8ab^2 \][/tex]
The denominator is:
[tex]\[ a^2b \cdot (3a - 65) = a^3b (3a - 65) \][/tex]
4. Divide the terms:
Rewrite the expression by dividing each term in the numerator by the corresponding term in the denominator:
[tex]\[ \frac{4a^2b - 8ab^2}{a^3b(3a - 65)} = \frac{4a^2b}{a^3b(3a - 65)} - \frac{8ab^2}{a^3b(3a - 65)} \][/tex]
Simplifying each term separately:
[tex]\[ \frac{4a^2b}{a^3b(3a - 65)} = \frac{4a^2b}{a^3b * (3a - 65)} = \frac{4a^2}{a^3 (3a - 65)} = \frac{4}{a (3a - 65)} \][/tex]
[tex]\[ \frac{8ab^2}{a^3b(3a - 65)} = \frac{8ab^2}{a^3b (3a - 65)} = \frac{8b}{a^2 (3a - 65)} \][/tex]
5. Further simplification:
Simplifying the combined expression:
[tex]\[ \frac{4a^2b - 8ab^2}{a^3b (3a - 65)} = \frac{4a (a - 2b)}{a^2 (3a - 65)} = \frac{4(a - 2b)}{a (3a - 65)} \][/tex]
So, the simplified form of the expression is:
[tex]\[ \frac{4(a - 2b)}{a(3a - 65)} \][/tex]
Therefore, the final answers are:
1. The solutions to [tex]\( x^2 - 8x + 3 = 0 \)[/tex] are [tex]\( x \approx 7.6 \)[/tex] and [tex]\( x \approx 0.4 \)[/tex].
2. The simplified form of [tex]\(\frac{ab}{3a-65} \times \frac{4a-8b}{a^2b}\)[/tex] is [tex]\(\frac{4(a - 2b)}{a(3a - 65)}\)[/tex].
