Completing Square 40E0A0

1. **State the problem:** Solve the quadratic equation $$19x^2 + 380x + 1899 = 0$$ by completing the square. 2. **Divide the entire equation by the coefficient of $$x^2$$ to make it 1:** $$\frac{19x^2}{19} + \frac{380x}{19} + \frac{1899}{19} = 0$$ which simplifies to $$x^2 + 20x + 99.947 = 0$$ 3. **Isolate the constant term:** $$x^2 + 20x = -99.947$$ 4. **Complete the square:** Take half of the coefficient of $$x$$, which is $$20$$, half is $$10$$, then square it: $$10^2 = 100$$. Add and subtract $$100$$ on the left side: $$x^2 + 20x + 100 - 100 = -99.947$$ 5. **Rewrite the equation:** $$ (x + 10)^2 - 100 = -99.947$$ 6. **Move the constant term to the right side:** $$ (x + 10)^2 = -99.947 + 100$$ $$ (x + 10)^2 = 0.053$$ 7. **Take the square root of both sides:** $$ x + 10 = \pm \sqrt{0.053}$$ $$ x + 10 = \pm 0.230$$ 8. **Solve for $$x$$:** $$ x = -10 \pm 0.230$$ 9. **Write the two solutions:** $$ x_1 = -10 + 0.230 = -9.770$$ $$ x_2 = -10 - 0.230 = -10.230$$ **Final answer:** $$ x_1 = -9.770 \quad \text{or} \quad x_2 = -10.230$$