4x² + 42x - 72 = 0 - United Radiology

["4x² + 42x - 72 = 0: Complete Guide to Solving the Quadratic Equation", "Solving quadratic equations is a fundamental skill in algebra, crucial for students, mathematicians, and professionals across sciences and engineering. One such equation frequently encountered is 4x² + 42x - 72 = 0. This article simplifies the solving process, explains key concepts, and provides practical steps and formulas to help you master this quadratic equation effortlessly.", "---", "### Understanding the Equation: 4x² + 42x - 72 = 0", "The given equation is a standard quadratic equation in the form:
\nax² + bx + c = 0, where
\n- a = 4
\n- b = 42
\n- c = -72", "To solve for x, we apply well-established algebraic methods such as factoring, completing the square, or using the quadratic formula. Each method offers unique insight into the nature of the roots.", "---", "### Step 1: Simplify the Equation (if Possible)", "Before jumping into complex formulas, simplify your equation if possible. Dividing every term by the greatest common divisor (GCD) of 4, 42, and -72, which is 2:", "2x² + 21x - 36 = 0", "This simplified form makes calculations cleaner and reduces the risk of arithmetic errors.", "---", "### Step 2: Choose a Solving Method", "There are three primary methods to solve 4x² + 42x - 72 = 0. We’ll explore each in detail.", "#### Method 1: Using the Quadratic Formula", "The quadratic formula is reliable and widely applicable:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "For our simplified equation 2x² + 21x - 36 = 0,
\na = 2, b = 21, c = -36", "Calculate discriminant (D):
\n[
\nD = b^2 - 4ac = 21^2 - 4(2)(-36) = 441 + 288 = 729
\n]", "Since D = 729, which is a perfect square (√729 = 27), the equation has two distinct rational roots.", "Now substitute into the formula:", "[
\nx = \frac{-21 \pm 27}{2 \cdot 2} = \frac{-21 \pm 27}{4}
\n]", "Calculate both roots:", "- First root:
\n[
\nx_1 = \frac{-21 + 27}{4} = \frac{6}{4} = \frac{3}{2} = 1.5
\n]", "- Second root:
\n[
\nx_2 = \frac{-21 - 27}{4} = \frac{-48}{4} = -12
\n]", "✅ Solutions:
\nx = -12 or x = 1.5", "---", "#### Method 2: Factoring (When Possible)", "Factoring works well when the quadratic can be expressed as a product of binomials. Start with:", "2x² + 21x - 36 = 0", "Looking for two numbers that multiply to (2)(-36) = -72 and add to 21.", "Such numbers are 24 and -3:
\n- 24 × (-3) = -72
\n- 24 + (-3) = 21", "Rewrite the middle term:", "[
\n2x² + 24x - 3x - 36 = 0
\n]", "Group terms:", "[
\n(2x² + 24x) + (-3x - 36) = 0
\n]
\n[
\n2x(x + 12) - 3(x + 12) = 0
\n]
\n[
\n(2x - 3)(x + 12) = 0
\n]", "Set each factor to zero:", "- 2x - 3 = 0 → x = 3/2 = 1.5
\n- x + 12 = 0 → x = -12", "Same solutions confirm our earlier result.", "---", "#### Method 3: Completing the Square (Conceptual Understanding)", "Completing the square is a powerful algebraic technique useful for understanding parabolas and maximizing/minimizing functions.", "Start with:", "[
\n2x² + 21x - 36 = 0
\n]", "Divide everything by 2 to simplify:", "[
\nx² + \frac{21}{2}x - 18 = 0
\n]", "Move constant to the right:", "[
\nx² + \frac{21}{2}x = 18
\n]", "Take half the coefficient of x → (21/4), square it → (441/16):", "Add (441/16) to both sides:", "[
\nx² + \frac{21}{2}x + \frac{441}{16} = 18 + \frac{441}{16}
\n]
\n[
\n\left(x + \frac{21}{4}\right)^2 = \frac{288}{16} + \frac{441}{16} = \frac{729}{16}
\n]", "Take square roots:", "[
\nx + \frac{21}{4} = \pm \frac{27}{4}
\n]", "Solve:", "- x = -21/4 + 27/4 = 6/4 = 1.5
\n- x = -21/4 - 27/4 = -48/4 = -12", "Again, confirming consistent results.", "---", "### Understanding the Roots and Graph", "The solutions x = -12 and x = 1.5 represent the x-intercepts of the parabola described by y = 4x² + 42x - 72. Since the leading coefficient (a = 4) is positive, the parabola opens upward, and the roots indicate where the graph crosses the x-axis.", "---", "### Real-World Applications", "Quadratic equations like 4x² + 42x - 72 = 0 model real-life scenarios such as:", "- Projectile motion trajectories
\n- Revenue and cost optimization in economics
\n- Electrical circuit analysis
\n- Geometry problems involving areas and dimensions", "Understanding their solutions empowers problem-solving across disciplines.", "---", "### Summary", "Solving 4x² + 42x - 72 = 0 can be managed efficiently via:
\n- Simplifying the equation to 2x² + 21x - 36 = 0
\n- Using the quadratic formula with discriminant D = 729
\n- Factoring by finding multiplicative pairs
\n- Conceptually completing the square", "Mastery of these methods not only solves specific equations but builds a strong foundation for advanced mathematics and practical applications.", "---", "### Further Learning: Want to Deepen Your Understanding?", "- Explore how the discriminant determines root nature (real, complex, repeated)
\n- Practice solving other quadratics by completing the square
\n- Use graphing calculators or software to visualize the function
\n- Apply quadratic equations in physics, economics, or engineering projects", "---", "Keywords for SEO: quadratic equation 4x² + 42x - 72, solve 4x² + 42x - 72, quadratic formula step-by-step, factoring quadratic, completing the square, solve 2x² + 21x - 36, discriminant meaning, real roots of quadratic, algebra practice problems.", "---", "Mastering quadratic equations opens new doors in mathematics and science—practice regularly, and soon you’ll solve them with confidence and ease."]