#### \( x^2 - 8x + 15 = 0 \)

["# Solving ( x^2 - 8x + 15 = 0 ): A Complete Guide for Students and Math Enthusiasts", "## Introduction", "Quadratic equations form a cornerstone of algebra, and solving ( x^2 - 8x + 15 = 0 ) is a fundamental task that helps build strong mathematical skills. Whether you're preparing for school exams, tackling standardized tests, or simply exploring algebra, understanding how to solve this equation step-by-step is essential. This comprehensive guide walks you through solving the quadratic equation ( x^2 - 8x + 15 = 0 ) using multiple methods, explains key concepts, and highlights common pitfalls to avoid.", "---", "## Understanding the Equation", "The equation ( x^2 - 8x + 15 = 0 ) is a standard form quadratic equation of the form:", "[\nax^2 + bx + c = 0\n]", "Here, the coefficients are:\n- ( a = 1 )\n- ( b = -8 )\n- ( c = 15 )", "Quadratic equations can be solved using several techniques: factoring, the quadratic formula, and completing the square. For this equation, factoring is straightforward and efficient.", "---", "## Method 1: Factoring", "### Step-by-step Factorization", "1. Look for two numbers that multiply to ( c = 15 ) and add up to ( b = -8 ).", "The numbers ( -3 ) and ( -5 ):\n - Multiply: ( (-3) \ imes (-5) = 15 )\n - Add: ( (-3) + (-5) = -8 )", "2. Write the factored form using these numbers:", "[\nx^2 - 8x + 15 = (x - 3)(x - 5) = 0\n]", "3. Apply the zero product property: If a product of factors is zero, then at least one factor must be zero.", "[\nx - 3 = 0 \quad \ ext{or} \quad x - 5 = 0\n]", "4. Solve each equation:", "[\nx = 3 \quad \ ext{or} \quad x = 5\n]", "---", "## Verifying the Roots", "Substitute ( x = 3 ):", "[\n(3)^2 - 8(3) + 15 = 9 - 24 + 15 = 0 \quad \ ext{✓}\n]", "Substitute ( x = 5 ):", "[\n(5)^2 - 8(5) + 15 = 25 - 40 + 15 = 0 \quad \ ext{✓}\n]", "Both solutions are correct.", "---", "## Method 2: Using the Quadratic Formula", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute ( a = 1 ), ( b = -8 ), ( c = 15 ):", "[\nx = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(15)}}{2(1)} = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm \sqrt{4}}{2} = \frac{8 \pm 2}{2}\n]", "So:", "[\nx = \frac{8 + 2}{2} = 5 \quad \ ext{and} \quad x = \frac{8 - 2}{2} = 3\n]", "Confirmed: ( x = 3 ) and ( x = 5 ).", "---", "## Method 3: Completing the Square", "Rewrite the equation:", "[\nx^2 - 8x + 15 = 0\n]", "Move the constant:", "[\nx^2 - 8x = -15\n]", "Take half of the coefficient of ( x ), square it:\n( (-8/2)^2 = 16 )", "Add 16 to both sides:", "[\nx^2 - 8x + 16 = 1\n]", "Factor the left side:", "[\n(x - 4)^2 = 1\n]", "Take square roots:", "[\nx - 4 = \pm 1 \Rightarrow x = 4 \pm 1\n]", "So the solutions are:", "[\nx = 5 \quad \ ext{or} \quad x = 3\n]", "Consistent with previous results.", "---", "## Why Solving ( x^2 - 8x + 15 = 0 ) Matters", "- Mastery of factoring: Recognizing factor pairs helps solve similar quadratics quickly.\n- Understanding roots: Knowing that each root satisfies the equation strengthens comprehension of graph behavior.\n- Foundation for advanced topics: Quadratics underpin concepts in calculus, physics, engineering, and economics.", "---", "## Tips to Avoid Common Mistakes", "- Double-check arithmetic in factoring and calculations.\n- Use the zero product property carefully—ensure both factors are solved.\n- Verify results by plugging back into the original equation.\n- Understand the discriminant ( b^2 - 4ac ) to predict root types (e.g., positive → two real roots).", "---", "## Summary", "Solving ( x^2 - 8x + 15 = 0 ) reveals two elegant real roots: ( x = 3 ) and ( x = 5 ). Whether using factoring, the quadratic formula, or completing the square, the solutions remain consistent. This equation exemplifies the beauty and utility of algebra—foundational, practical, and indispensable.", "---", "### Want to practice more?", "Try solving related equations:\n( x^2 - 5x + 6 = 0 ),\n( x^2 + 2x - 8 = 0 ),\nor explore real-world applications like projectile motion modeled by quadratics.", "---", "Keywords: ( x^2 - 8x + 15 = 0 ), quadratic equation solutions, factoring quadratics, quadratic formula, zero product property, algebra practice, math tutorial, solving quadratics, real roots, factoring techniques, discriminant, quadratic roots.", "---", "Meta description: Clear step-by-step solution for ( x^2 - 8x + 15 = 0 ) using factoring, quadratic formula, and completing the square. Learn algebra basics with verified methods and practical verification."]