Question**: Solve the quadratic equation \( 2x^2 - 5x - 3 = 0 \) using the quadratic formula.
["# Solve the Quadratic Equation ( 2x^2 - 5x - 3 = 0 ) Using the Quadratic Formula", "Trying to solve a quadratic equation like ( 2x^2 - 5x - 3 = 0 )? Understanding how to apply the quadratic formula is a key skill in algebra. Whether you're a student, educator, or math enthusiast, mastering this technique ensures you can tackle any standard quadratic equation with confidence. In this article, we’ll walk through step-by-step instructions on solving ( 2x^2 - 5x - 3 = 0 ) using the quadratic formula, along with explanations and verified results.", "---", "## What is the Quadratic Equation?", "A quadratic equation is any equation in the standard form:", "[\nax^2 + bx + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ). The quadratic formula provides a direct way to find the solutions (or roots) of any quadratic equation:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula works universally, regardless of whether the roots are real or complex.", "---", "## Step-by-Step: Solving ( 2x^2 - 5x - 3 = 0 )", "### Step 1: Identify coefficients ( a ), ( b ), and ( c )", "From the equation ( 2x^2 - 5x - 3 = 0 ), compare it with the standard form:", "- ( a = 2 )\n- ( b = -5 )\n- ( c = -3 )", "### Step 2: Compute the discriminant ( \Delta )", "The discriminant determines the nature of the roots:", "[\n\Delta = b^2 - 4ac\n]", "Substitute the values:", "[\n\Delta = (-5)^2 - 4(2)(-3) = 25 + 24 = 49\n]", "Since ( \Delta > 0 ), there are two distinct real roots.", "### Step 3: Plug values into the quadratic formula", "[\nx = \frac{-(-5) \pm \sqrt{49}}{2 \cdot 2}\n]", "Simplify:", "[\nx = \frac{5 \pm 7}{4}\n]", "### Step 4: Calculate both solutions", "[\nx_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3\n]", "[\nx_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}\n]", "---", "## Final Answer", "The solutions to the quadratic equation ( 2x^2 - 5x - 3 = 0 ) are:", "[\nx = 3 \quad \ ext{and} \quad x = -\frac{1}{2}\n]", "---", "## Why This Method Works", "Using the quadratic formula ensures accuracy and applies to all quadratic equations—regardless of how the coefficients are set up. This systematic approach eliminates trial and error, making it ideal for exams, homework, and real-life problem solving.", "---", "## Practice Tips", "- Always compute the discriminant first to know the type of roots.\n- Simplify fractions carefully when applying the formula.\n- Double-check your arithmetic at each step to avoid sign errors.", "---", "## Conclusion", "Solving ( 2x^2 - 5x - 3 = 0 ) with the quadratic formula is straightforward once you understand the formula and its components. By following the steps above, you’ll confidently solve any quadratic equation and strengthen your algebra foundation. Remember: consistent practice with diverse problems leads to mastery!", "---", "Keywords: quadratic formula, solve quadratic equation, quadratic formula steps, solve (2x^2 - 5x - 3 = 0), real roots of quadratic, algebraic solutions, algebra practice, discriminant calculation."]
