4x^2 + 70x - 350 = 0

["Solving the Quadratic Equation 4x² + 70x – 350 = 0: Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, widely used in science, engineering, economics, and everyday problem-solving. This article explores the equation 4x² + 70x – 350 = 0, breaking down how to find its roots using the quadratic formula, and explains the mathematical concepts behind it. Whether you're a student, educator, or self-learner, understanding how to solve such equations opens doors to deeper analytical thinking and practical applications.", "---", "### Understanding the Quadratic Equation", "A quadratic equation has the standard form:", "[\nax^2 + bx + c = 0\n]", "where:\n- ( a ) is the coefficient of ( x^2 ),\n- ( b ) is the coefficient of ( x ),\n- ( c ) is the constant term.", "For the equation 4x² + 70x – 350 = 0, we identify:\n- ( a = 4 )\n- ( b = 70 )\n- ( c = -350 )", "---", "### Step 1: Use the Quadratic Formula", "The most reliable method to solve any quadratic equation is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute ( a = 4 ), ( b = 70 ), and ( c = -350 ) into the formula.", "---", "### Step 2: Calculate the Discriminant", "First, compute the discriminant ( D ), which determines the nature of the roots:", "[\nD = b^2 - 4ac\n]", "[\nD = (70)^2 - 4(4)(-350) = 4900 + 5600 = 10,500\n]", "Since ( D > 0 ), the equation has two distinct real roots.", "---", "### Step 3: Find the Square Root of the Discriminant", "Now calculate ( \sqrt{D} = \sqrt{10,500} ). Simplify it:", "[\n\sqrt{10,500} = \sqrt{100 \ imes 105} = 10\sqrt{105}\n]", "---", "### Step 4: Substitute Values into the Quadratic Formula", "Plug values back into the formula:", "[\nx = \frac{-70 \pm 10\sqrt{105}}{2 \cdot 4} = \frac{-70 \pm 10\sqrt{105}}{8}\n]", "Simplify numerator and denominator:", "[\nx = \frac{-35 \pm 5\sqrt{105}}{4}\n]", "---", "### Step 5: Final Solutions", "The two real solutions are:", "[\nx_1 = \frac{-35 + 5\sqrt{105}}{4}, \quad x_2 = \frac{-35 - 5\sqrt{105}}{4}\n]", "---", "### Step 6: Approximate Numerical Values (Optional)", "For practical use, approximate ( \sqrt{105} \approx 10.247 ):", "[\nx_1 \approx \frac{-35 + 5(10.247)}{4} = \frac{-35 + 51.235}{4} = \frac{16.235}{4} \approx 4.06\n]", "[\nx_2 \approx \frac{-35 - 51.235}{4} = \frac{-86.235}{4} \approx -21.56\n]", "---", "### Why This Equation Matters", "The roots of 4x² + 70x – 350 = 0 represent solution points where the parabola defined by the function ( f(x) = 4x^2 + 70x - 350 ) crosses the x-axis. Understanding these points is valuable in optimization, physics, and economics—where quadratic models describe motion, profit, or physical systems.", "---", "### Practice Tips for Solving Quadratics", "- Always simplify coefficients before applying the quadratic formula.\n- Carefully compute the discriminant to anticipate root types (real, repeated, or complex).\n- Use factoring or completing the square only when convenient; the quadratic formula is universally reliable.\n- Whenever possible, verify roots by substituting them back into the original equation.", "---", "### Summary", "Solving 4x² + 70x – 350 = 0 yields two real roots:", "[\nx = \frac{-35 \pm 5\sqrt{105}}{4}\n]", "These solutions illustrate the power of algebraic methods in uncovering precise mathematical results. Whether for homework, exams, or real-world modeling, mastering quadratic equations strengthens critical thinking and problem-solving abilities.", "---", "Keywords:\nquadratic equation, 4x² + 70x – 350 = 0, solve quadratic, quadratic formula, discriminant, real roots, algebra tutorial, solving equations, square root simplification, mathematical solutions.", "---", "Need more help with quadratics? Explore our guides on factoring, completing the square, or real-world applications in projectile motion and business profit analysis. Keep learning — the math world opens up with each equation!"]