Solving Quadratics by Factoring $$ 7.3 x^{2}-21=4-x^{2} $$

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1. **Write the original equation:**

$$
   7.3x^2 - 21 = 4 - x^2
   $$

2. **Bring all terms to one side:**

Add $$x^2$$ to both sides and subtract $$4$$ from both sides:

$$
   7.3x^2 + x^2 - 21 - 4 = 0
   $$

Combine like terms:

$$
   8.3x^2 - 25 = 0
   $$

3. **Clear the decimal by multiplying the entire equation by 10:**

$$
   10(8.3x^2 - 25) = 10(0)
   $$
   
   This gives:

$$
   83x^2 - 250 = 0
   $$

4. **Express the equation in the form of a difference of squares:**

Write $$83x^2$$ and $$250$$ as squares. Notice that:

$$
   83x^2 = \left(\sqrt{83}\,x\right)^2 \quad \text{and} \quad 250 = \left(5\sqrt{10}\right)^2
   $$
   
   Thus, the equation becomes:

$$
   \left(\sqrt{83}\, x\right)^2 - \left(5\sqrt{10}\right)^2 = 0
   $$

5. **Factor the difference of squares:**

Using the identity $$a^2 - b^2 = (a-b)(a+b)$$:

$$
   \left(\sqrt{83}\,x - 5\sqrt{10}\right)\left(\sqrt{83}\,x + 5\sqrt{10}\right) = 0
   $$

6. **Set each factor equal to zero and solve for $$x$$:**

- For the first factor:

$$
     \sqrt{83}\,x - 5\sqrt{10} = 0 \quad \Rightarrow \quad \sqrt{83}\,x = 5\sqrt{10} \quad \Rightarrow \quad x = \frac{5\sqrt{10}}{\sqrt{83}}
     $$

- For the second factor:

$$
     \sqrt{83}\,x + 5\sqrt{10} = 0 \quad \Rightarrow \quad \sqrt{83}\,x = -5\sqrt{10} \quad \Rightarrow \quad x = -\frac{5\sqrt{10}}{\sqrt{83}}
     $$

7. **Present the solutions:**

$$
   x = \frac{5\sqrt{10}}{\sqrt{83}} \quad \text{or} \quad x = -\frac{5\sqrt{10}}{\sqrt{83}}
   $$
The solutions are $$ x = \frac{5\sqrt{10}}{\sqrt{83}} $$ and $$ x = -\frac{5\sqrt{10}}{\sqrt{83}} $$.

Solving Quadratics by Factoring \( 7.3 x^{2}-21=4-x^{2} \)

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