["# Solving the Quadratic Equation: ( 2x^2 + 2x - 84 = 0 )", "Quadratic equations are a fundamental part of algebra and play a crucial role in various scientific and engineering applications. One commonly encountered equation is ( 2x^2 + 2x - 84 = 0 ). In this article, we will provide a clear, step-by-step solution to this equation, explain how to solve it using different methods, and highlight its practical relevance.", "---", "## What is the Equation ( 2x^2 + 2x - 84 = 0 )?", "This is a standard quadratic equation of the form ( ax^2 + bx + c = 0 ), where:
\n- ( a = 2 )
\n- ( b = 2 )
\n- ( c = -84 )", "Solving such equations helps us find the values of ( x ) (also called roots) that satisfy the relationship.", "---", "## Step-by-Step Solution", "### Method 1: Using the Quadratic Formula", "The most general method to solve ( ax^2 + bx + c = 0 ) is the quadratic formula:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Step 1: Identify coefficients
\n( a = 2 ), ( b = 2 ), ( c = -84 )", "Step 2: Calculate the discriminant ( D = b^2 - 4ac )", "[
\nD = (2)^2 - 4(2)(-84) = 4 + 672 = 676
\n]", "The discriminant is positive, so there are two distinct real roots.", "Step 3: Plug values into the quadratic formula:", "[
\nx = \frac{-2 \pm \sqrt{676}}{2 \cdot 2} = \frac{-2 \pm 26}{4}
\n]", "Step 4: Calculate the two solutions:", "[
\nx_1 = \frac{-2 + 26}{4} = \frac{24}{4} = 6
\n]", "[
\nx_2 = \frac{-2 - 26}{4} = \frac{-28}{4} = -7
\n]", "---", "### Method 2: Factoring", "After dividing the equation by 2 (to simplify):", "[
\nx^2 + x - 42 = 0
\n]", "We look for two numbers that multiply to ( -42 ) and add to ( 1 ). Those numbers are ( 7 ) and ( -6 ):", "[
\n(x + 7)(x - 6) = 0
\n]", "Setting each factor equal to zero:", "[
\nx + 7 = 0 \Rightarrow x = -7
\n]
\n[
\nx - 6 = 0 \Rightarrow x = 6
\n]", "This matches the results from the quadratic formula.", "---", "## Key Takeaways", "- The equation ( 2x^2 + 2x - 84 = 0 ) simplifies to ( x^2 + x - 42 = 0 ), which factors neatly.
\n- The solutions are ( x = 6 ) and ( x = -7 ).
\n- The discriminant (( 676 )) confirms two real and distinct roots.
\n- Using the quadratic formula guarantees a correct and efficient solution.", "---", "## Why Solve quadratic equations like ( 2x^2 + 2x - 84 = 0 )?", "- They model real-world phenomena such as projectile motion and profit maximization.
\n- Finding roots helps in optimization and decision-making in business and science.
\n- Mastery of solving quadratics strengthens algebraic foundations necessary for higher math.", "---", "## Practice Trouble-Free", "Try solving this yourself:", "- Estimate where the parabola crosses the x-axis.
\n- Use the quadratic formula on ( 2x^2 + 2x - 84 = 0 ) and verify your answer.", "---", "## Conclusion", "The equation ( 2x^2 + 2x - 84 = 0 ) is a classic quadratic that illustrates key algebraic techniques. Whether solved via factoring or the quadratic formula, the roots ( x = 6 ) and ( x = -7 ) demonstrate the power of algebra to uncover critical values. Understanding how to solve such equations enhances your problem-solving skills and supports advanced study in mathematics and its applications.", "---", "Keywords: ( 2x^2 + 2x - 84 = 0 ), quadratic equation, solve quadratic, quadratic formula, discriminant, factoring, algebra solutions, real roots, quadratic applications", "Meta Description: Learn how to solve ( 2x^2 + 2x - 84 = 0 ) step-by-step using the quadratic formula and factoring, including solution values ( x = 6 ) and ( x = -7 ). Ideal for students and math enthusiasts."]
