The problem asks to solve the quadratic equation $2x^2 = 4x + 3$ using the quadratic formula.

Step-by-Step Solution:

1. Rearrange into Standard Quadratic Form: The standard quadratic form is $ax^2 + bx + c = 0$. Rearrange $2x^2 = 4x + 3$: $2x^2 - 4x - 3 = 0$

2. Identify Coefficients: From $2x^2 - 4x - 3 = 0$, the coefficients are: $a = 2, \quad b = -4, \quad c = -3$

3. Quadratic Formula: The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

4. Substitute Coefficients into the Formula: Substitute $a = 2$, $b = -4$, $c = -3$: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-3)}}{2(2)}$ Simplify step-by-step: $x = \frac{4 \pm \sqrt{16 - (-24)}}{4}$ $x = \frac{4 \pm \sqrt{16 + 24}}{4}$ $x = \frac{4 \pm \sqrt{40}}{4}$

5. Simplify the Square Root: The square root of 40 can be simplified: $\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$ So: $x = \frac{4 \pm 2\sqrt{10}}{4}$

6. Simplify the Expression: Divide both terms in the numerator by 4: $x = \frac{4}{4} \pm \frac{2\sqrt{10}}{4}$ $x = 1 \pm \frac{\sqrt{10}}{2}$

Final Answer:

The solutions are: $x = 1 + \frac{\sqrt{10}}{2}, \quad x = 1 - \frac{\sqrt{10}}{2}$

Would you like me to explain any step in more detail or verify your calculations?

Related Questions:

1. How do you derive the quadratic formula?
2. Can you explain why the discriminant ($b^2 - 4ac$) determines the nature of solutions?
3. How do you simplify expressions involving square roots?
4. What are the applications of quadratic equations in real-life scenarios?
5. Can this quadratic equation be solved by factoring instead of the quadratic formula?

Tip:

Always double-check your discriminant ($b^2 - 4ac$) before solving, as it helps verify whether the solutions will be real or complex!