Quadratic Inequality 30E473

1. **State the problem:** Solve the quadratic inequality $$4x^2 + 8x - 21 > 0$$ and represent the solution set graphically. 2. **Find the roots of the quadratic equation:** To solve the inequality, first find the roots of the equation $$4x^2 + 8x - 21 = 0$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=4$$, $$b=8$$, and $$c=-21$$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 8^2 - 4 \times 4 \times (-21) = 64 + 336 = 400$$ 4. **Calculate the roots:** $$x = \frac{-8 \pm \sqrt{400}}{2 \times 4} = \frac{-8 \pm 20}{8}$$ So, $$x_1 = \frac{-8 - 20}{8} = \frac{-28}{8} = -\frac{7}{2} = -3.5$$ $$x_2 = \frac{-8 + 20}{8} = \frac{12}{8} = \frac{3}{2} = 1.5$$ 5. **Rewrite roots as decimals for clarity:** $$x_1 = -3.5$$ $$x_2 = 1.5$$ 6. **Analyze the inequality:** Since the leading coefficient $$a=4 > 0$$, the parabola opens upwards. The quadratic expression is positive outside the roots and negative between them. 7. **Solution set:** $$x < -3.5 \quad \text{or} \quad x > 1.5$$ 8. **Graphical representation:** On the number line, mark points $$A$$ at $$x = -3.5$$ and $$B$$ at $$x = 1.5$$. Shade the regions to the left of $$A$$ and to the right of $$B$$ to represent the solution set. **Final answer:** $$\boxed{x < -3.5 \text{ or } x > 1.5}$$