Mathematics 10 — Exercise 5.4 · Q1 (v)

Full question

Solve the following equation by reducing it to a quadratic equation. Check the resulting values in the original equation.

Step-by-step solution

1. Use the exponent laws to write the equation as a quadratic in t, where t is the following positive power.

   $$t=4^x$$

2. After substitution the quadratic equation is:

   $$4\times t^2-10\times t+4=0$$

3. Substitute the function expressions, keeping the brackets.

   $$4\times t^2-10\times t+4-0$$

4. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$4\times t^2-10\times t+4$$

5. Write EVERY distributed product before collecting terms.

   $$4-10\times t+4\times t^2-0$$

6. Calculate this individual product, including its sign and powers.

   $$4=4$$

7. Calculate this individual product, including its sign and powers.

   $$-10\times t=-10\times t$$

8. Calculate this individual product, including its sign and powers.

   $$4\times t^2=4\times t^2$$

9. Calculate this individual product, including its sign and powers.

   $$\left(-1\right)\times 0=0$$

10. These terms have the same variable part. Add ONLY their coefficients.

    $$4+0=4$$

11. These terms have the same variable part. Add ONLY their coefficients.

    $$-10=-10$$

12. These terms have the same variable part. Add ONLY their coefficients.

    $$4=4$$

13. Put the simplified groups together.

    $$4\times t^2-10\times t+4$$

14. Collect terms with the same power and simplify their coefficients.

    $$4\times t^2-10\times t+4$$

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

    $$4\times t^2-10\times t+4=0$$

16. Split the middle coefficient into two numbers whose sum is b and whose product is a·c.

    $$(-2-8=-10,\left(-2\right)\times \left(-8\right)=16)$$

17. Rewrite the middle term using those two parts.

    $$4\times t^2-2\times t-8\times t+4=0$$

18. Group the first two terms and the last two. Take out the common factor from each group.

    $$\left(2\times t\right)\times \left(2\times t-1\right)-4\times \left(2\times t-1\right)=0$$

19. The common bracket can now be factored out.

    $$2\times \left(t-2\right)\times \left(2\times t-1\right)=0$$

20. A product is zero only if at least one factor is zero. Set each linear factor equal to zero.

    $$(t-2=0,2\times t-1=0)$$

21. Take this factor equal to zero.

    $$t-2=0$$

22. Subtract the constant from both sides.

    $$t-2+2=0+2$$

23. Simplify both sides.

    $$t=2$$

24. Divide both sides by the coefficient of the variable.

    $$t=2\times \frac{1}{1}$$

25. Reduce the fraction to obtain this root.

    $$t=2$$

26. Take this factor equal to zero.

    $$2\times t-1=0$$

27. Subtract the constant from both sides.

    $$2\times t-1+1=0+1$$

28. Simplify both sides.

    $$2\times t=1$$

29. Divide both sides by the coefficient of the variable.

    $$t=\frac{1}{2}$$

30. Reduce the fraction to obtain this root.

    $$t=\frac{1}{2}$$

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

    $$0=0$$

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

    $$0=0$$

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

    $$0=0$$

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

    $$0=0$$

35. Set the original power equal to this positive t value.

    $$4^x=\frac{1}{2}$$

36. Write both sides with the same base if possible; otherwise use a logarithm.

    $$x=-\frac{\log \left(2\right)}{\log \left(4\right)}$$

37. Divide the exponent equation by the multiplier of x.

    $$x=-\frac{1}{2}$$

38. Set the original power equal to this positive t value.

    $$4^x=2$$

39. Write both sides with the same base if possible; otherwise use a logarithm.

    $$x=\frac{\log \left(2\right)}{\log \left(4\right)}$$

40. Divide the exponent equation by the multiplier of x.

    $$x=\frac{1}{2}$$

41. Collect every valid root from every substitution branch.

    $$\{-\frac{1}{2},\frac{1}{2}\}$$

Answer

The final answer is shown below.