Question: c) 3 - x^2 = -5x + 7
1. **State the problem:** Solve the quadratic equation $$3 - x^2 = -5x + 7$$.
2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:
$$3 - x^2 + 5x - 7 = 0$$
3. **Simplify:**
$$-x^2 + 5x - 4 = 0$$
4. **Multiply both sides by -1** to make the leading coefficient positive:
$$\cancel{-1} \times (-x^2 + 5x - 4) = \cancel{-1} \times 0$$
$$x^2 - 5x + 4 = 0$$
5. **Use the quadratic formula:** For equation $$ax^2 + bx + c = 0$$, the solutions are
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
where $$a=1$$, $$b=-5$$, and $$c=4$$.
6. **Calculate the discriminant:**
$$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 4 = 25 - 16 = 9$$
7. **Find the roots:**
$$x = \frac{-(-5) \pm \sqrt{9}}{2 \times 1} = \frac{5 \pm 3}{2}$$
8. **Calculate each root:**
- $$x_1 = \frac{5 + 3}{2} = \frac{8}{2} = 4$$
- $$x_2 = \frac{5 - 3}{2} = \frac{2}{2} = 1$$
**Final answer:** The solutions to the equation are $$x = 4$$ and $$x = 1$$.