["# Solving the Quadratic Equation x² + 2x − 15 = 0: A Step-by-Step Guide", "If you're diving into algebra or preparing for a math exam, understanding how to solve quadratic equations is essential. One frequently encountered problem is the quadratic equation:
\nx² + 2x − 15 = 0", "In this comprehensive guide, we’ll walk through solving this equation using multiple methods—factoring, completing the square, and the quadratic formula—while explaining the key concepts and verifying your solutions. Whether you're a student, teacher, or self-learner, mastering this equation strengthens your foundational math skills.", "---", "## What Is a Quadratic Equation?", "A quadratic equation is any equation of the form:
\nax² + bx + c = 0, where a ≠ 0. The graph of a quadratic equation is a parabola, and its solutions (roots) tell us where the parabola intersects the x-axis.", "For our example:
\nx² + 2x − 15 = 0
\nHere, a = 1, b = 2, and c = −15.", "---", "## Method 1: Factoring", "Factoring is often the fastest way to solve quadratics, especially when the equation has integer solutions.", "### Step 1: Look for two numbers
\nWe need two numbers that:
\n- Multiply to c = –15
\n- Add to b = +2", "After checking factor pairs of −15 (like 5 and −3, or −5 and 3), we find:
\n5 + (–3) = 2 and 5 × (–3) = –15", "### Step 2: Rewrite the equation
\nUsing these numbers, factor the quadratic:
\nx² + 2x − 15 = (x + 5)(x − 3) = 0", "### Step 3: Apply the Zero Product Property
\nIf (x + 5)(x − 3) = 0, then:
\nx + 5 = 0 → x = –5
\nx − 3 = 0 → x = 3", "✅ Final solutions: x = –5 and x = 3", "---", "## Method 2: Quadratic Formula", "While factoring works well here, the quadratic formula is a universal tool for any q ∈ ℝ:
\n$$
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n$$", "Plug in a = 1, b = 2, c = –15:
\n$$
\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(–15)}}{2(1)}
\n= \frac{-2 \pm \sqrt{4 + 60}}{2}
\n= \frac{-2 \pm \sqrt{64}}{2}
\n= \frac{-2 \pm 8}{2}
\n$$", "Now calculate both solutions:
\n1. $ x = \frac{–2 + 8}{2} = \frac{6}{2} = 3 $
\n2. $ x = \frac{–2 – 8}{2} = \frac{–10}{2} = –5 $", "✅ Same solutions: x = –5 and x = 3", "---", "## Method 3: Completing the Square", "This algebraic technique transforms the equation into a perfect square trinomial, making it easy to solve.", "### Step 1: Move the constant
\nx² + 2x = 15", "### Step 2: Complete the square
\nTake half of b (which is 2), square it:
\n(2/2)² = 1² = 1
\nAdd 1 to both sides:
\nx² + 2x + 1 = 15 + 1 → (x + 1)² = 16", "### Step 3: Take square roots
\nx + 1 = ±√16 → x + 1 = ±4
\nSo:
\nx = –1 + 4 = 3
\nx = –1 – 4 = –5", "✅ Confirmed solutions: x = 3 and x = –5", "---", "## Why Is This Equation Important?", "- It illustrates key algebraic techniques used in more advanced math.
\n- Quadratic equations model real-world phenomena like projectile motion and optimization problems.
\n- Practicing this equation helps build confidence with factoring, skip counting (factors), and square roots.", "---", "## Summary", "The quadratic equation x² + 2x – 15 = 0 can be solved efficiently using:", "- Factoring: (x + 5)(x − 3) = 0 → x = –5, 3
\n- Quadratic Formula: x = –5, 3
\n- Completing the Square: (x + 1)² = 16 → x = 3, –5", "Understanding multiple solutions reinforces flexibility with algebra and prepares you for factoring more complex quadratics.", "---", "## Need More Practice?", "Try these related problems:
\n- Solve: x² + 2x + 8 = 0 (shh—complex roots!)
\n- Graph y = x² + 2x – 15
\n- Apply quadratics in physics: projectile height = –5t² + 20t + 10", "---", "## FAQ: Frequently Asked Questions", "Q: What does it mean if the equation has no real solutions?
\nA: If the discriminant (b² – 4ac) is negative, there are no real roots—just complex ones.", "Q: Can x² + 2x – 15 factored with decimals?
\nA: Yes, but factoring with integers is preferred for clarity. For example, if c were −12, (x + 6)(x – 2) = 0.", "Q: How do I verify my solutions?
\nA: Plug x = 3 and x = –5 back into the original equation—both satisfy x² + 2x – 15 = 0.", "---", "#quadraticequation #algebra #solveequation #factoring #quadraticformula #mathtips #algebra101 #problem-solving"]
