["# Simplifying the Quadratic Equation (4x^2 - 50x + 40 = 0) Step by Step", "Solving quadratic equations is a fundamental skill in algebra, and while full simplification and factoring can sometimes be straightforward, using efficient techniques helps reduce complexity quickly. In this article, we explore how to further simplify the quadratic equation:", "[
\n4x^2 - 50x + 40 = 0
\n]", "## Step 1: Simplify Coefficients by Factoring Out the Greatest Common Factor (GCF)", "The first step in simplifying a quadratic equation is to factor out the greatest common factor (GCF) from all terms. This makes the equation easier to work with and reduces potential computational errors.", "Given:", "[
\n4x^2 - 50x + 40
\n]", "Check the coefficients:
\n- 4 = 2 × 2
\n- 50 = 2 × 5 × 5
\n- 40 = 2 × 2 × 2 × 5", "The GCF is 2. Factor it out:", "[
\n2(2x^2 - 25x + 20) = 0
\n]", "Now the simplified form is:", "[
\n2x^2 - 25x + 20 = 0
\n]", "Since 2 is a constant multiplier, if required for further solving (like using the quadratic formula), you can divide the entire equation by 2:", "[
\nx^2 - \frac{25}{2}x + 10 = 0
\n]", "However, working with integers remains simpler in most cases, so we often keep it as:", "[
\n2x^2 - 25x + 20 = 0
\n]", "## Step 2: Simplify Factoring — Use the AC Method", "Now we attempt to factor the simplified quadratic expression:", "[
\n2x^2 - 25x + 20
\n]", "AC method steps:", "- Multiply (a \cdot c = 2 \ imes 20 = 40)
\n- Find two numbers that multiply to 40 and add to (-25).
\n- Those numbers are -20 and -5, because:", "[
\n-20 \ imes -5 = 100 \quad \ ext{(Not 40 — correction needed!)}
\n]", "Wait — actually, we need two numbers multiplying to (2 \ imes 20 = 40), and adding to (-25), but these integers don’t multiply cleanly to 40 and add to -25.", "Instead, we proceed with factoring by grouping, the most reliable method for this case.", "Rewrite the middle term using split coefficients:", "We want to write:", "[
\n2x^2 - 25x + 20 = 0
\n]", "We look for two binomials of the form:", "[
\n(ax + m)(bx + n) = 0
\n]", "such that:
\n- (a \cdot b = 2)
\n- (m \cdot n = 20)
\n- (an + bm = -25)", "Try (a = 2), (b = 1) (since 2 × 1 = 2):", "We need:", "[
\n(2x + m)(x + n) = 2x^2 + (2n + m)x + mn
\n]", "So:
\n- (mn = 20)
\n- (2n + m = -25)", "Try integer pairs for (m) and (n) multiplying to 20:", "- ((-1, -20)): (2(-20) + (-1) = -40 -1 = -41)
\n- ((-2, -10)): (2(-10) + (-2) = -20 - 2 = -22)
\n- ((-4, -5)): (2(-5) + (-4) = -10 - 4 = -14)
\n- ((-5, -4)): (2(-4) + (-5) = -8 - 5 = -13)
\n- ((-20, -1)): already tried", "None give -25. Try switching factors: suppose the quadratic factors as:", "[
\n(2x - 5)(x - 4) = 2x^2 - 8x - 5x + 20 = 2x^2 - 13x + 20 \quad \ ext{(Too low)}
\n]", "Try:", "[
\n(2x - 20)(x - 1) = 2x^2 - 2x - 20x + 20 = 2x^2 - 22x + 20 \quad \ ext{Too low}
\n]", "Try:", "[
\n(2x - 4)(x - 5) = 2x^2 - 10x - 4x + 20 = 2x^2 - 14x + 20
\n]", "Still not matching.", "Try:", "[
\n(2x - 1)(x - 20) = 2x^2 - 40x - x + 20 = 2x^2 - 41x + 20
\n]", "No.", "After checking, this quadratic does not factor nicely over integers. Thus, we proceed to use the quadratic formula, the most robust method.", "## Step 3: Solve Using the Quadratic Formula (Best Practice When Factoring Is Hard)", "Given simplified equation:", "[
\n2x^2 - 25x + 20 = 0
\n]", "Use the quadratic formula:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Identify coefficients:
\n(a = 2), (b = -25), (c = 20)", "Calculate discriminant:", "[
\n\Delta = (-25)^2 - 4(2)(20) = 625 - 160 = 465
\n]", "Since 465 is not a perfect square ((\sqrt{465} \approx 21.56)), the solutions are irrational.", "Now compute:", "[
\nx = \frac{25 \pm \sqrt{465}}{4}
\n]", "Thus, the two solutions are:", "[
\nx = \frac{25 + \sqrt{465}}{4} \quad \ ext{and} \quad x = \frac{25 - \sqrt{465}}{4}
\n]", "## Summary: Simplified Solving Pathway", "- Factored out GCF 2: (2(2x^2 - 25x + 20) = 0)
\n- Confirmed equation does not factor neatly with integers
\n- Applied quadratic formula for exact solution:
\n [
\n x = \frac{25 \pm \sqrt{465}}{4}
\n ]", "While full numerical simplification isn’t elegant, this is the simplified exact algebraic solution. Simplifying by factoring led us to a form where the quadratic formula applies directly — the most efficient approach once factoring proves unworkable.", "---", "Key Takeaways for Further Simplification:", "1. Always factor out the GCF — reduces complexity.
\n2. When coefficients are large or non-integer, consider using the quadratic formula.
\n3. Though not factorable over integers, breaking down via grouping or trial may still offer insights — but the equation resolves cleanest via:
\n [
\n x = \frac{25 \pm \sqrt{465}}{4}
\n ]
\n4. For real-world use, entering this form into calculators or graphing tools yields precise solutions efficiently.", "---", "By mastering GCF extraction and quick assessment of factorability, any quadratic equation can be simplified effectively — turning complex expressions into usable solutions.", "Keywords:
\nquadratic equation simplification, solve (4x^2 - 50x + 40 = 0), factor quadratic, quadratic formula, simplify (2x^2 - 25x + 20 = 0), exact solutions, Discriminant (\Delta = 465), algebraic simplification techniques", "---", "Optimize your learning: practice simplifying different quadratics to recognize factorable patterns quickly and know when to apply the quadratic formula."]
