1. **State the problem:** Solve the quadratic equation $$4x^2 - x - 3 = 0$$ using an algebraic method. 2. **Formula and method:** We use the quadratic formula to solve equations of the form $$ax^2 + bx + c = 0$$, which is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=4$, $b=-1$, and $c=-3$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-1)^2 - 4 \times 4 \times (-3) = 1 + 48 = 49$$ Since $\Delta > 0$, there are two real solutions. 4. **Apply the quadratic formula:** $$x = \frac{-(-1) \pm \sqrt{49}}{2 \times 4} = \frac{1 \pm 7}{8}$$ 5. **Find the two solutions:** - For the plus sign: $$x = \frac{1 + 7}{8} = \frac{8}{8} = 1$$ - For the minus sign: $$x = \frac{1 - 7}{8} = \frac{-6}{8} = -\frac{3}{4}$$ 6. **Final answer:** The solutions to the equation are $$x = 1$$ and $$x = -\frac{3}{4}$$. **Graph shape and math description:** The graph of $$y = 4x^2 - x - 3$$ is a parabola opening upwards (since $a=4 > 0$). It crosses the x-axis at the points $x=1$ and $x=-\frac{3}{4}$, which are the roots we found.