["Simplifying Quadratic Equations: A Step-by-Step Guide to Solving (4x^2 - 50x + 150 = 110)", "When solving quadratic equations, simplifying the equation properly is key to efficiently finding the correct solutions. In this article, we’ll walk through the process of simplifying and solving:
\n[
\n4x^2 - 50x + 150 = 110
\n]", "### Step 1: Move All Terms to One Side
\nThe first step in simplifying any quadratic equation is to bring all terms to one side, forming a standard form:
\n[
\nax^2 + bx + c = 0
\n]
\nSubtract 110 from both sides:
\n[
\n4x^2 - 50x + 150 - 110 = 0
\n]
\n[
\n4x^2 - 50x + 40 = 0
\n]", "### Step 2: Simplify the Equation
\nNext, check if the quadratic can be simplified by factoring out a common divisor. Here, all coefficients share a common factor of 2:
\n[
\n2(2x^2 - 25x + 20) = 0
\n]
\nDividing both sides by 2 gives:
\n[
\n2x^2 - 25x + 20 = 0
\n]
\nNow, the equation is simpler and easier to solve.", "### Step 3: Use the Quadratic Formula
\nThis simplified quadratic equation is in the form:
\n[
\nax^2 + bx + c = 0
\n]
\nwhere ( a = 2 ), ( b = -25 ), and ( c = 20 ).", "The quadratic formula is:
\n[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Plugging in the values:
\n[
\nx = \frac{-(-25) \pm \sqrt{(-25)^2 - 4(2)(20)}}{2(2)}
\n]
\n[
\nx = \frac{25 \pm \sqrt{625 - 160}}{4}
\n]
\n[
\nx = \frac{25 \pm \sqrt{465}}{4}
\n]", "### Step 4: Final Answer
\nThe simplified solutions are:
\n[
\nx = \frac{25 \pm \sqrt{465}}{4}
\n]
\nThis is the simplest exact form. For approximate decimal values, use a calculator:
\n[
\nx \approx \frac{25 \pm 21.56}{4}
\n]
\nSo,
\n[
\nx \approx \frac{46.56}{4} = 11.64 \quad \ ext{or} \quad x \approx \frac{3.44}{4} = 0.86
\n]", "---", "Summary: Simplifying quadratic equations—by rearranging terms and factoring out common coefficients—makes solving easier. Using the quadratic formula with simplified coefficients helps avoid calculation errors and leads to clean, accurate solutions. Mastering these techniques ensures confidence in tackling more complex equations.", "Keywords: simplify quadratic equation, solve (4x^2 - 50x + 150 = 110), quadratic formula, simplify (ax^2 + bx + c = 0), step-by-step quadratic simplification, exact quadratic solutions.", "---", "Related Topics:
\n- How to use the quadratic formula effectively
\n- Factors of quadratic equations with common divisors
\n- Step-by-step: From standard form to solution"]
