Simplify: \(4x^2 + 70x - 350 = 0\).

["# Simplifying the Quadratic Equation: (4x^2 + 70x - 350 = 0)", "Solving quadratic equations is a fundamental skill in algebra, and simplifying the equation is the first step toward finding its roots efficiently. In this article, we walk through how to simplify and solve the equation (4x^2 + 70x - 350 = 0) using clear, step-by-step methods suitable for learners and educators alike.", "---", "## Understanding the Equation", "We begin with the standard quadratic form:\n[\nax^2 + bx + c = 0\n]\nFor the equation (4x^2 + 70x - 350 = 0), the coefficients are:\n- (a = 4)\n- (b = 70)\n- (c = -350)", "---", "## Step 1: Simplify the Equation", "Before jumping into complex solving techniques, simplifying the equation makes calculations easier.", "### Factoring out the GCD", "First, notice that all three terms are divisible by 2:\n[\n4x^2 + 70x - 350 = 0\n\Rightarrow 2(2x^2 + 35x - 175) = 0\n]", "Divide both sides by 2:\n[\n2x^2 + 35x - 175 = 0\n]", "Now, the equation is simplified, with smaller coefficients and easier handling. This step is crucial because working with whole-number coefficients reduces mistakes and improves clarity when applying factoring or the quadratic formula.", "---", "## Step 2: Factoring the Quadratic (If Possible)", "We attempt to factor (2x^2 + 35x - 175).", "Look for two numbers that:\n- Multiply to (2 \ imes (-175) = -350)\n- Add up to (35)", "After testing divisor pairs of -350, we find:\n(50 \ imes (-7) = -350) and (50 + (-7) = 43) → not correct\n(70 \ imes (-5) = -350), but (70 - 5 = 65) → not quite\nFinally, (50) and (-7) don’t work, but trying (35) and (-10) doesn’t help either.", "Since no convenient factor pairing exists quickly, factoring by grouping may not be straightforward, so we shift to the most reliable method for such quadratics: the quadratic formula.", "---", "## Step 3: Apply the Quadratic Formula", "The quadratic formula solves (ax^2 + bx + c = 0) via:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute (a = 2), (b = 35), (c = -175):", "[\nx = \frac{-35 \pm \sqrt{(35)^2 - 4(2)(-175)}}{2(2)}\n]", "Calculate discriminant:\n[\nb^2 - 4ac = 1225 + 1400 = 2625\n]", "Now, simplify (\sqrt{2625}):\nFactor out perfect squares:\n(2625 = 25 \ imes 105 = 25 \ imes (3 \ imes 5 \ imes 7) = 5^2 \ imes 105), so\n[\n\sqrt{2625} = 5\sqrt{105}\n]", "(Note: 105 has no square factors beyond 1, so this radical cannot be simplified further.)", "Now plug back:\n[\nx = \frac{-35 \pm 5\sqrt{105}}{4}\n]", "---", "## Step 4: Final Simplified Solutions", "Thus, the simplified exact solutions are:\n[\nx = \frac{-35 + 5\sqrt{105}}{4} \quad \ ext{and} \quad x = \frac{-35 - 5\sqrt{105}}{4}\n]", "These represent the two real roots of the equation. Since the discriminant is positive, we have two distinct real solutions.", "---", "## Why Simplifying Matters", "Simplifying equations like (4x^2 + 70x - 350 = 0) improves:\n- Ease of factoring\n- Accuracy in calculations\n- Clarity when plugging into formulas\n- Test-taking and writing efficiency", "---", "## Conclusion", "Simplifying (4x^2 + 70x - 350 = 0) by factoring out 2 before factoring made the solution process clearer and more manageable. While this equation doesn’t factor easily into integers, applying the quadratic formula with simplified coefficients ensures accuracy and mastery.", "Whether you're preparing for exams or mastering algebra fundamentals, understanding how to simplify and solve quadratics is essential. Keep practicing—each equation brings you closer to algebraic fluency!", "---", "### Related Topics:\n- How to solve quadratic equations by factoring\n- Simplifying quadratic expressions\n- Understanding the quadratic formula and discriminant\n- Real-world applications of solving quadratics", "Visit our algebra hub for more tools to simplify equations and boost your math skills!"]