Answer
Here are the solutions to the quadratic equations:
6. \( t = \frac{-3 \pm \sqrt{69}}{10} \)
7. \( t = 2 \pm \sqrt{2} \)
8. \( k = \frac{7 \pm \sqrt{373}}{18} \)
9. \( f = \frac{1}{2} \) or \( f = -2 \)
10. \( t = \frac{-1 \pm i\sqrt{3}}{2} \)
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Solution
Below are complete solutions for each quadratic:
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Problem 6. Solve 5t² + 3t – 3 = 0.
Step 1. Identify the coefficients:
a = 5, b = 3, c = –3.
Step 2. Use the quadratic formula:
t = [–b ± √(b² – 4ac)]/(2a).
Step 3. Compute the discriminant:
b² – 4ac = 3² – 4(5)(–3) = 9 + 60 = 69.
Step 4. Write the solutions:
t = [–3 ± √69] / (2·5) = (–3 ± √69)/10.
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Problem 7. Solve t² – 4t + 2 = 0.
Step 1. Identify the coefficients:
a = 1, b = –4, c = 2.
Step 2. Use the quadratic formula:
t = [–(–4) ± √((–4)² – 4(1)(2)]/2 = [4 ± √(16 – 8)]/2.
Step 3. Simplify the discriminant:
16 – 8 = 8 and √8 = 2√2.
Step 4. Write the solutions:
t = (4 ± 2√2)/2 = 2 ± √2.
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Problem 8. Solve 9(k² – 1) = 7k.
Step 1. Expand the left side:
9k² – 9 = 7k.
Step 2. Rearrange to standard form:
9k² – 7k – 9 = 0.
Step 3. Identify the coefficients:
a = 9, b = –7, c = –9.
Step 4. Use the quadratic formula:
k = [–(–7) ± √((–7)² – 4(9)(–9)]/(2·9) = [7 ± √(49 + 324)]/18.
Step 5. Simplify the discriminant:
49 + 324 = 373.
Step 6. Write the solutions:
k = (7 ± √373) / 18.
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Problem 9. Solve 3f – 2 = –2f².
Step 1. Rearrange the equation:
Add 2f² to both sides: 2f² + 3f – 2 = 0.
Step 2. Identify the coefficients:
a = 2, b = 3, c = –2.
Step 3. Use the quadratic formula:
f = [–3 ± √(3² – 4(2)(–2)]/(2·2) = [–3 ± √(9 + 16)]/4.
Step 4. Simplify the discriminant:
9 + 16 = 25 and √25 = 5.
Step 5. Write the solutions:
f = (–3 ± 5)/4.
This gives:
f = (–3 + 5)/4 = 2/4 = ½ or f = (–3 – 5)/4 = –8/4 = –2.
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Problem 10. Solve t² + t + 1 = 0.
Step 1. Identify the coefficients:
a = 1, b = 1, c = 1.
Step 2. Compute the discriminant:
b² – 4ac = 1 – 4 = –3.
Step 3. Since the discriminant is negative, the solutions are complex:
t = [–1 ± √(–3)]/2 = (–1 ± i√3)/2.
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Summary of Answers:
6. t = (–3 ± √69)/10
7. t = 2 ± √2
8. k = (7 ± √373)/18
9. f = ½ or f = –2
10. t = (–1 ± i√3)/2
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