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

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:** \[ x = \frac{5\sqrt{10}}{\sqrt{83}} \quad \text{or} \quad x = -\frac{5\sqrt{10}}{\sqrt{83}} \]