\[ x^2 + 2x - 15 = 120 \] - United Radiology

["# Solving the Quadratic Equation: x² + 2x - 15 = 120", "Mastering quadratic equations is essential for students, math enthusiasts, and professionals working in fields like physics, engineering, or economics. One common quadratic problem students regularly encounter is solving equations such as:", "x² + 2x - 15 = 120", "Understanding how to simplify, rearrange, and solve this equation unlocks strong algebra skills and prepares you for more advanced math concepts. This article provides a step-by-step guide to solving x² + 2x - 15 = 120, explains key algebraic principles, and offers useful tips for finding solutions quickly and accurately.", "---", "## Step 1: Rearrange the Equation to Standard Form", "The first step in solving any quadratic equation is putting it into standard form:", "[
\nax² + bx + c = 0
\n]", "Starting with:", "[
\nx² + 2x - 15 = 120
\n]", "Subtract 120 from both sides:", "[
\nx² + 2x - 15 - 120 = 0
\n]", "[
\nx² + 2x - 135 = 0
\n]", "Now your equation is in standard quadratic form:", "[
\nx² + 2x - 135 = 0
\n]", "---", "## Step 2: Apply the Quadratic Formula", "For equations of the form (ax² + bx + c = 0), the quadratic formula gives the solutions:", "[
\nx = \frac{-b \pm \sqrt{b² - 4ac}}{2a}
\n]", "For x² + 2x - 135 = 0, the coefficients are:", "- (a = 1)
\n- (b = 2)
\n- (c = -135)", "Substitute into the formula:", "[
\nx = \frac{-2 \pm \sqrt{(2)² - 4(1)(-135)}}{2(1)}
\n]", "[
\nx = \frac{-2 \pm \sqrt{4 + 540}}{2}
\n]", "[
\nx = \frac{-2 \pm \sqrt{544}}{2}
\n]", "Now simplify (\sqrt{544}):", "[
\n\sqrt{544} = \sqrt{16 \ imes 34} = 4\sqrt{34}
\n]", "So,", "[
\nx = \frac{-2 \pm 4\sqrt{34}}{2}
\n]", "Simplify by dividing numerator terms by 2:", "[
\nx = -1 \pm 2\sqrt{34}
\n]", "---", "## Step 3: Final Solutions", "The two real solutions are:", "[
\nx = -1 + 2\sqrt{34} \quad \ ext{and} \quad x = -1 - 2\sqrt{34}
\n]", "### Approximate numeric values (for reference):", "[
\n\sqrt{34} \approx 5.831
\n]", "So,", "[
\nx \approx -1 + 2(5.831) = -1 + 11.662 = 10.662
\n]", "[
\nx \approx -1 - 11.662 = -12.662
\n]", "---", "## Why Understanding This Equation Matters", "Solving x² + 2x - 15 = 120 is more than just finding roots—it builds foundational algebra skills:", "- Rearranging equations helps isolate variables in real-world contexts like profit analysis or distance-time calculations.
\n- Quadratic formulas are key in physics (projectile motion), business (revenue optimization), and computer science (graphics rendering).
\n- Working with irrational numbers improves comfort with radicals and enhances numerical reasoning.", "---", "## Tips for Quick Solving", "- Move constants first: Always simplify right-hand side before factoring or applying formulas.
\n- Check solutions: Plug values back into the original equation to confirm accuracy.
\n- Use exact vs. decimal: Providing answers in radical form (like ( -1 \pm 2\sqrt{34} )) is preferred in academic settings.
\n- Practice with variations: Try equations like (x² + bx + c = \ ext{constant not zero}) to strengthen pattern recognition.", "---", "## Conclusion", "The equation (x² + 2x - 15 = 120) may appear simple, but mastering its solution reveals powerful algebraic techniques. By rearranging, applying the quadratic formula, and understanding irrational solutions, you gain tools essential for advanced math and practical problem-solving. Whether you're preparing for school exams or tackling real-world math challenges, steady practice with quadratics builds confidence and competence.", "---", "Ready to solve more? Explore our guides on quadratic inequalities, completing the square, and applications of polynomials—mastery starts here!", "---", "Keywords:
\nx² + 2x - 15 = 120, solving quadratic equations, quadratic formula, algebra tutorial, solving x² + 2x - 135 = 0, step-by-step quadratic solutions, real and irrational roots, math help quadratic equations"]