\[ x^2 + 2x - 288 = 0 \] - United Radiology

["# Solving the Quadratic Equation: ( x^2 + 2x - 288 = 0 )", "Understanding how to solve quadratic equations is a fundamental skill in algebra and essential for many real-world applications. One commonly encountered equation is ( x^2 + 2x - 288 = 0 ). In this comprehensive guide, we’ll explore step-by-step how to solve this quadratic equation, uncover its roots, and discuss practical uses.", "## What is the Equation ( x^2 + 2x - 288 = 0 )?", "This is a standard quadratic equation in the form ( ax^2 + bx + c = 0 ), where:
\n- ( a = 1 )
\n- ( b = 2 )
\n- ( c = -288 )", "Quadratic equations arise in various fields—from physics and engineering to economics—and their solutions give precise values of ( x ) that satisfy the equation.", "## Step-by-Step Solution", "### 1. Identify coefficients
\nFrom ( x^2 + 2x - 288 = 0 ):
\n- ( a = 1 )
\n- ( b = 2 )
\n- ( c = -288 )", "### 2. Use the Quadratic Formula
\nThe most reliable method to solve any quadratic equation is the quadratic formula:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Substitute ( a = 1 ), ( b = 2 ), and ( c = -288 ) into the formula:", "[
\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-288)}}{2(1)}
\n]", "### 3. Calculate the discriminant
\nThe discriminant ( D = b^2 - 4ac ) determines the nature of the roots:", "[
\nD = 2^2 - 4(1)(-288) = 4 + 1152 = 1156
\n]", "Since ( D > 0 ), there are two distinct real solutions.", "### 4. Evaluate the square root
\n[
\n\sqrt{1156} = 34
\n]", "### 5. Substitute back and simplify
\n[
\nx = \frac{-2 \pm 34}{2}
\n]", "Now calculate both roots:", "- First root:
\n[
\nx = \frac{-2 + 34}{2} = \frac{32}{2} = 16
\n]", "- Second root:
\n[
\nx = \frac{-2 - 34}{2} = \frac{-36}{2} = -18
\n]", "### 6. Final Solution
\nThe solutions to ( x^2 + 2x - 288 = 0 ) are:", "[
\nx = 16 \quad \ ext{and} \quad x = -18
\n]", "## Verification", "To confirm correctness, plug ( x = 16 ) and ( x = -18 ) back into the original equation:", "- For ( x = 16 ):
\n( (16)^2 + 2(16) - 288 = 256 + 32 - 288 = 0 )", "- For ( x = -18 ):
\n( (-18)^2 + 2(-18) - 288 = 324 - 36 - 288 = 0 )", "Both values satisfy the equation.", "## Why Knowing These Roots Matters", "- Word problems: These roots can represent critical points in optimization problems.
\n- Graphing: Roots indicate where the parabola ( y = x^2 + 2x - 288 ) intersects the x-axis.
\n- Real-world applications: Used in physics for projectile motion and in finance for break-even analysis.", "## Alternative Methods", "Besides the quadratic formula, you can factor the equation if possible. Factoring ( x^2 + 2x - 288 ):
\nWe seek two numbers that multiply to ( -288 ) and add to ( 2 ). These numbers are ( 18 ) and ( -16 ):", "[
\nx^2 + 18x - 16x - 288 = 0
\nx(x + 18) -16(x + 18) = 0
\n(x - 16)(x + 18) = 0
\n]", "Setting each factor to zero confirms the roots:", "[
\nx - 16 = 0 \Rightarrow x = 16
\nx + 18 = 0 \Rightarrow x = -18
\n]", "## Summary", "The equation ( x^2 + 2x - 288 = 0 ) has two real solutions:
\n( x = 16 ) and ( x = -18 ).
\nUsing the quadratic formula and understanding the discriminant allows precise and reliable solutions. Mastering these techniques empowers problem-solving in academics and everyday life.", "## SEO Keywords
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