1. **State the problem:** Solve the quadratic equation $$15x^2 + x - 6 = 0$$ by factorization and find $x$. 2. **Factorize the quadratic:** Looking for two numbers that multiply to $$15 \times (-6) = -90$$ and add to $$1$$. These numbers are $$10$$ and $$-9$$. 3. **Rewrite the middle term:** $$15x^2 + 10x - 9x - 6 = 0$$ 4. **Group terms:** $$(15x^2 + 10x) - (9x + 6) = 0$$ 5. **Factor each group:** $$5x(3x + 2) - 3(3x + 2) = 0$$ 6. **Factor out the common binomial:** $$(5x - 3)(3x + 2) = 0$$ 7. **Set each factor equal to zero:** $$5x - 3 = 0 \Rightarrow x = \frac{3}{5}$$ $$3x + 2 = 0 \Rightarrow x = -\frac{2}{3}$$ --- 1. **State the problem:** Solve the quadratic equation $$x^2 + 6x + 4 = 0$$ by factorization and find $x$. 2. **Use quadratic formula:** Since factorization isn't straightforward, use $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=6$$, $$c=4$$. 3. **Calculate discriminant:** $$\Delta = 6^2 - 4 \times 1 \times 4 = 36 - 16 = 20$$ 4. **Calculate solutions:** $$x = \frac{-6 \pm \sqrt{20}}{2} = \frac{-6 \pm 2\sqrt{5}}{2} = -3 \pm \sqrt{5}$$ 5. **Final roots:** $$x_1 = -3 + \sqrt{5}$$ and $$x_2 = -3 - \sqrt{5}$$ --- 1. **State the problem:** Solve the quadratic equation $$x^2 - 4x - 14 = 0$$ by factorization and find $x$. 2. **Use quadratic formula:** Here, $$a=1$$, $$b=-4$$, $$c=-14$$. 3. **Calculate discriminant:** $$\Delta = (-4)^2 - 4 \times 1 \times (-14) = 16 + 56 = 72$$ 4. **Calculate solutions:** $$x = \frac{4 \pm \sqrt{72}}{2} = \frac{4 \pm 6\sqrt{2}}{2} = 2 \pm 3\sqrt{2}$$ 5. **Final roots:** $$x_1 = 2 + 3\sqrt{2}$$ and $$x_2 = 2 - 3\sqrt{2}$$