1. **State the problem:** Solve the quadratic equation $4x^2 - 8x = -3$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$4x^2 - 8x + 3 = 0$$ 3. **Identify coefficients:** Here, $a=4$, $b=-8$, and $c=3$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$b^2 - 4ac = (-8)^2 - 4 \times 4 \times 3 = 64 - 48 = 16$$ 6. **Calculate the roots:** $$x = \frac{-(-8) \pm \sqrt{16}}{2 \times 4} = \frac{8 \pm 4}{8}$$ 7. **Find each solution:** - For $+$ sign: $$x = \frac{8 + 4}{8} = \frac{12}{8} = \frac{\cancel{12}}{\cancel{8}} = \frac{3}{2}$$ - For $-$ sign: $$x = \frac{8 - 4}{8} = \frac{4}{8} = \frac{\cancel{4}}{\cancel{8}} = \frac{1}{2}$$ **Final answer:** $$x = \frac{3}{2} \text{ or } x = \frac{1}{2}$$