QUESTION 3 Determine, WITHOUT the use of a calculator, the value of $$ x $$ in the follow 3.1. $$ (x-5)(3+x)=0 $$ 3.2. $$ \frac{x^{2}-3}{2}=x $$

Question

UpStudy AI · Accepted Answer

**3.1. Solve $$ (x-5)(3+x)=0 $$**

Since the product of two factors is zero, set each factor equal to zero:

- $$ x-5=0 $$ ⟹ $$ x=5 $$
- $$ 3+x=0 $$ ⟹ $$ x=-3 $$

Thus, the solutions are $$ x=5 $$ and $$ x=-3 $$.

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**3.2. Solve $$ \frac{x^{2}-3}{2}=x $$**

1. Multiply both sides by 2 to eliminate the fraction:
   $$
   x^2 - 3 = 2x
   $$

2. Rearrange the equation into standard quadratic form:
   $$
   x^2 - 2x - 3 = 0
   $$

3. Factor the quadratic:
   $$
   (x-3)(x+1) = 0
   $$

4. Set each factor equal to zero:
   - $$ x-3=0 $$ ⟹ $$ x=3 $$
   - $$ x+1=0 $$ ⟹ $$ x=-1 $$

Thus, the solutions are $$ x=3 $$ and $$ x=-1 $$.
**3.1.** $$ x = 5 $$ or $$ x = -3 $$ **3.2.** $$ x = 3 $$ or $$ x = -1 $$

QUESTION 3 Determine, WITHOUT the use of a calculator, the value of \( x \) in the follow 3.1. \( (x-5)(3+x)=0 \) 3.2. \( \frac{x^{2}-3}{2}=x \)