Mathematics 10 — Exercise 2.7 · Q3

Full question

A toy car travels s = t² − 0.5t metres after t seconds. When has it travelled 12.5 metres?

Step-by-step solution

1. A denominator cannot equal zero. Keep the original excluded values throughout the calculation.

   $$2\ne 0$$

2. Multiply BOTH sides by the common denominator to remove the fractions.

   $$2\times \left(t^2-\frac{t}{2}\right)=2\times \frac{25}{2}$$

3. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

   $$2\times t^2-t-25=0$$

4. Calculate this operation before combining the remaining terms.

   $$8\times \left(-25\right)=-200$$

5. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times \left(-200\right)=200$$

6. Compute the discriminant one operation at a time: square b, multiply 4,a,c, then subtract.

   $${\left(-1\right)}^2-8\times \left(-25\right)$$

7. Calculate this operation before combining the remaining terms.

   $${\left(-1\right)}^2=1$$

8. Calculate this operation before combining the remaining terms.

   $$\left(-4\right)\times 2\times \left(-25\right)=200$$

9. Calculate this operation before combining the remaining terms.

   $$1+200=201$$

10. Compare with ax² + bx + c = 0 and read a, b, c WITH their signs.

    $$(a=2,b=-1,c=-25)$$

11. Compute D = b² − 4ac. A minus times a minus becomes a plus.

    $$D={\left(-1\right)}^2-8\times \left(-25\right)$$

12. Simplify the discriminant.

    $$D=201$$

13. Use x = (−b ± √D)/(2a). Show the plus and minus cases separately.

    $$(t=\frac{1+\sqrt{201}}{4},t=\frac{1-\sqrt{201}}{4})$$

14. Simplify each numerator and divide by 2a.

    $$\{\frac{1}{4}-\frac{1}{4}\times \sqrt{201},\frac{1}{4}+\frac{1}{4}\times \sqrt{201}\}$$

15. Calculate this root branch separately, preserving the sign.

    $$t=\frac{1+1\times \sqrt{201}}{2\times 2}$$

16. Calculate this operation before combining the remaining terms.

    $$1\times \sqrt{201}=\sqrt{201}$$

17. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

18. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 4}$$

19. Calculate these two products separately.

    $$\frac{1}{4}$$

20. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{4}=\frac{1}{4}$$

21. Calculate this operation before combining the remaining terms.

    $$1+\sqrt{201}\times \frac{1}{4}=\frac{1}{4}+\frac{1}{4}\times \sqrt{201}$$

22. Calculate this root branch separately, preserving the sign.

    $$t=\frac{1-\sqrt{201}}{2\times 2}$$

23. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

24. Calculate this operation before combining the remaining terms.

    $$\left(1-\sqrt{201}\right)\times \frac{1}{4}=\frac{1}{4}-\frac{1}{4}\times \sqrt{201}$$

25. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 1}{1\times 8}$$

26. Calculate these two products separately.

    $$-\frac{1}{8}$$

27. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{8}=-\frac{1}{8}$$

28. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 8}$$

29. Calculate these two products separately.

    $$\frac{1}{8}$$

30. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{8}=\frac{1}{8}$$

31. Calculate this operation before combining the remaining terms.

    $${\left(\frac{1}{4}+\frac{1}{4}\times \sqrt{201}\right)}^2=\frac{1}{8}\times \sqrt{201}+\frac{101}{8}$$

32. Calculate this operation before combining the remaining terms.

    $$-\frac{1}{8}\times \sqrt{201}-\frac{1}{8}+\frac{1}{8}\times \sqrt{201}+\frac{101}{8}=\frac{25}{2}$$

33. Substitute this root into the original equation before simplifying.

    $$\left(-\frac{1}{8}\right)\times \sqrt{201}-\frac{1}{8}+{\left(\frac{1}{4}+\frac{1}{4}\times \sqrt{201}\right)}^2=\frac{25}{2}$$

34. Calculate both sides; their equal values confirm the root.

    $$\frac{25}{2}=\frac{25}{2}$$

35. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{4}\times \sqrt{201}=\left(-\frac{1}{4}\right)\times \sqrt{201}$$

36. Calculate this operation before combining the remaining terms.

    $${\left(\frac{1}{4}-\frac{1}{4}\times \sqrt{201}\right)}^2=\frac{101}{8}-\frac{1}{8}\times \sqrt{201}$$

37. Calculate this operation before combining the remaining terms.

    $$-\frac{1}{8}+\frac{1}{8}\times \sqrt{201}+\frac{101}{8}-\frac{1}{8}\times \sqrt{201}=\frac{25}{2}$$

38. Substitute this root into the original equation before simplifying.

    $$-\frac{1}{8}+\frac{1}{8}\times \sqrt{201}+{\left(\frac{1}{4}-\frac{1}{4}\times \sqrt{201}\right)}^2=\frac{25}{2}$$

39. Calculate both sides; their equal values confirm the root.

    $$\frac{25}{2}=\frac{25}{2}$$

40. Time cannot be negative. Keep the positive root only.

Answer

The final answer is shown below.