Mathematics 10 — Exercise 2.1 · Q4 (iv)

Full question

Use the quadratic formula to solve the following equation: x²+11 = 7 × x.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$x^2+11-7\times x$$

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

   $$x^2-7\times x+11$$

3. Write EVERY distributed product before collecting terms.

   $$11+x^2-7\times x$$

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

   $$11=11$$

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

   $$x^2=x^2$$

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

   $$-7\times x=-7\times x$$

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

   $$11=11$$

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

   $$1=1$$

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

   $$-7=-7$$

10. Put the simplified groups together.

    $$x^2-7\times x+11$$

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

    $$x^2-7\times x+11$$

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

    $$x^2-7\times x+11=0$$

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

    $${\left(-7\right)}^2-4\times 11$$

14. Calculate this operation before combining the remaining terms.

    $${\left(-7\right)}^2=49$$

15. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 1\times 11=-44$$

16. Calculate this operation before combining the remaining terms.

    $$49-44=5$$

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

    $$(a=1,b=-7,c=11)$$

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

    $$D={\left(-7\right)}^2-4\times 11$$

19. Simplify the discriminant.

    $$D=5$$

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

    $$(x=\frac{\sqrt{5}+7}{2},x=\frac{7-\sqrt{5}}{2})$$

21. Simplify each numerator and divide by 2a.

    $$\{\frac{7}{2}-\frac{1}{2}\times \sqrt{5},\frac{1}{2}\times \sqrt{5}+\frac{7}{2}\}$$

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

    $$x=\frac{1\times \sqrt{5}+7}{2\times 1}$$

23. Calculate this operation before combining the remaining terms.

    $$1\times \sqrt{5}=\sqrt{5}$$

24. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

25. Calculate this operation before combining the remaining terms.

    $$\sqrt{5}+7\times \frac{1}{2}=\frac{1}{2}\times \sqrt{5}+\frac{7}{2}$$

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

    $$x=\frac{7-\sqrt{5}}{2\times 1}$$

27. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

28. Calculate this operation before combining the remaining terms.

    $$7-\sqrt{5}\times \frac{1}{2}=\frac{7}{2}-\frac{1}{2}\times \sqrt{5}$$

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

    $$11+{\left(\frac{1}{2}\times \sqrt{5}+\frac{7}{2}\right)}^2=\frac{7}{2}\times \sqrt{5}+\frac{49}{2}$$

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

    $$\frac{7}{2}\times \sqrt{5}+\frac{49}{2}=\frac{7}{2}\times \sqrt{5}+\frac{49}{2}$$

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

    $${\left(\frac{7}{2}-\frac{1}{2}\times \sqrt{5}\right)}^2+11=\frac{49}{2}-\frac{7}{2}\times \sqrt{5}$$

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

    $$\frac{49}{2}-\frac{7}{2}\times \sqrt{5}=\frac{49}{2}-\frac{7}{2}\times \sqrt{5}$$

Answer

The final answer is shown below.