Solving this quadratic using the quadratic formula, w = [-70 ± √(70² - 4*4*(-96))] / (2*4).

["Solving a Quadratic Equation Using the Quadratic Formula: Step-by-Step Guide", "Quadratic equations appear frequently in algebra and have practical applications across science, engineering, and finance. If you're trying to solve a quadratic equation like ( w = [-70 ± √(70² - 4×4×(-96))] / (2×4) ), the quadratic formula is your most reliable tool. In this article, we’ll walk through the process of solving such equations using the quadratic formula with a clear breakdown of every step.", "---", "### What Is the Quadratic Formula?", "The standard form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "The quadratic formula solves for ( x ) using:", "[\nx = \frac{-b ± √(b² - 4ac)}{2a}\n]", "This formula works for any quadratic equation, regardless of whether it has real or complex solutions.", "---", "### Step 1: Identify ( a ), ( b ), and ( c )", "From your example:", "[\nw = \frac{-70 ± √(70² - 4×4×(-96))}{2×4}\n]", "Compare this with ( ax^2 + bx + c = 0 ) to identify coefficients:", "- ( a = 4 )\n- ( b = 70 )\n- ( c = -96 )", "(Note: The denominator ( 2×4 = 8 ) shows ( 2a), not just 2.)", "---", "### Step 2: Plug Values Into the Formula", "Start substituting values:", "[\nw = \frac{-70 ± √(70² - 4×4×(-96))}{2×4}\n]", "Break it down:", "- ( 70² = 4900 )\n- ( 4×4×(-96) = 16×(-96) = -1536 ) (negative times positive = negative, so subtract a negative becomes add)\n- ( b² - 4ac = 4900 - (-1536) = 4900 + 1536 = 6436 )", "So,", "[\nw = \frac{-70 ± √6436}{8}\n]", "---", "### Step 3: Simplify the Square Root (If Possible)", "Now simplify ( \sqrt{6436} ). See whether it can be simplified:", "Check if 6436 has perfect square factors. Try dividing:", "( 6436 ÷ 4 = 1609 ), and 1609 is a prime number (no small square factors).\nThus:", "[\n\sqrt{6436} = \sqrt{4×1609} = 2\sqrt{1609}\n]", "So the expression becomes:", "[\nw = \frac{-70 ± 2√1609}{8}\n]", "Simplify numerator and denominator by dividing by 2:", "[\nw = \frac{-35 ± √1609}{4}\n]", "---", "### Step 4: Write the Two Solutions", "Using the ± symbol, we get two solutions:", "[\nw_1 = \frac{-35 + √1609}{4}, \quad w_2 = \frac{-35 - √1609}{4}\n]", "These are the exact real solutions since the discriminant (( b² - 4ac = 6436 > 0 )) is positive.", "---", "### Optional: Decimal Approximation", "If you'd like decimal approximations:", "- ( √1609 ≈ 40.13 )", "Then:", "- ( w_1 ≈ \frac{-35 + 40.13}{4} = \frac{5.13}{4} ≈ 1.28 )\n- ( w_2 ≈ \frac{-35 - 40.13}{4} = \frac{-75.13}{4} ≈ -18.78 )", "---", "### Summary", "Solving quadratic equations using the quadratic formula is a powerful method:", "1. Identify ( a ), ( b ), and ( c ) from standard form.\n2. Substitute into the formula ( w = [-b ± √(b² - 4ac)] / (2a) ).\n3. Simplify the discriminant.\n4. Compute both real roots using ±.\n5. Simplify or approximate as needed.", "This method ensures accurate solutions even when factoring is difficult or impossible.", "---", "Key terms: quadratic formula, solve quadratic equation, simplify square root, algebra steps, quadratic formula example, solving ( w = [-70 ± √(70² - 4×4×(-96))]/(2×4) ), rational roots, irrational numbers."]