["Solve the Quadratic Equation ( 2x^2 + 35x - 102 = 0 ): A Step-by-Step Guide", "If you're tackling a quadratic equation like ( 2x^2 + 35x - 102 = 0 ), you're not alone—quadratic equations are fundamental in algebra and have wide-ranging applications in science, engineering, economics, and more. Mastering methods to solve them improves your problem-solving skills and deepens your understanding of polynomial functions.", "In this SEO-optimized article, we’ll walk you through solving ( 2x^2 + 35x - 102 = 0 ) clearly and efficiently, explaining both the factoring approach and the quadratic formula—two powerful techniques that guarantee accurate results.", "---", "### Why Solve Quadratic Equations Like This?", "- Foundational math skill: Quadratics are essential for understanding parabolas and real-world modeling.
\n- Application in various fields: Physics (motion), engineering (design), finance (profit maximization), and computer graphics (curve modeling).
\n- Boost analytical thinking: Solving ( ax^2 + bx + c = 0 ) enhances logical reasoning and algebraic manipulation.", "---", "### Step 1: Understand the Standard Form", "The general form of a quadratic equation is:
\n[
\nax^2 + bx + c = 0
\n]
\nFor our equation:
\n[
\n2x^2 + 35x - 102 = 0
\n]
\nHere, ( a = 2 ), ( b = 35 ), and ( c = -102 ).", "---", "### Step 2: Solve Using Factoring (When Possible)", "Factoring works best when the quadratic expression can be written as a product of two binomials. Let’s try to factor ( 2x^2 + 35x - 102 ).", "#### Guesswork & Trial
\nWe need two numbers that:
\n- Multiply to ( a \cdot c = 2 \ imes (-102) = -204 )
\n- Add up to ( b = 35 )", "After testing factor pairs of -204, we find:
\n( 49 \ imes (-4) = -196 ) ❌ (not enough)
\n( 54 \ imes (-4) = -216 ) ❌
\n( 51 \ imes (-4) = -204 ) ✔️ and ( 51 + (-4) = 47 ) ❌
\n( 54 \ imes (-4) = -216 ) ❌
\nEventually, we find:
\n( (2x + 51)(x - 2) = 0 ) → Check:
\n( 2x \cdot x = 2x^2 ), ( 2x \cdot (-2) = -4x ), ( 51x - 102 ) → Total: ( 2x^2 + 35x - 102 ) ✅", "So, the factored form is:
\n[
\n(2x + 51)(x - 2) = 0
\n]", "#### Solve Each Factor
\nSet each factor equal to zero:
\n( 2x + 51 = 0 \Rightarrow x = -\frac{51}{2} = -25.5 )
\n( x - 2 = 0 \Rightarrow x = 2 )", "> ✅ Solutions: ( x = -25.5 ) or ( x = 2 )", "---", "### Step 3: Verify Using the Quadratic Formula (When Factoring Is Hard)", "The quadratic formula is a reliable method for any ( ax^2 + bx + c = 0 ):
\n[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Plug in ( a = 2 ), ( b = 35 ), ( c = -102 ):
\n[
\nx = \frac{-35 \pm \sqrt{35^2 - 4(2)(-102)}}{2 \cdot 2}
\n]
\n[
\n= \frac{-35 \pm \sqrt{1225 + 816}}{4}
\n]
\n[
\n= \frac{-35 \pm \sqrt{2041}}{4}
\n]", "Since ( \sqrt{2041} ) is irrational, leave it as-is:
\n[
\nx = \frac{-35 \pm \sqrt{2041}}{4}
\n]", "Wait—this seems inconsistent with the factored solution. Why?", "> Important note: Earlier factoring missed perfect pairing. Let’s double-check. Trying the correct combination:
\nFactoring ( 2x^2 + 35x - 102 ), test factor pair ( (2x + 51)(x - 2) ) again:
\n( 2x \cdot x = 2x^2 ),
\n( 2x \cdot (-2) = -4x ),
\n( 51 \cdot x = 51x ),
\n( 51 \cdot (-2) = -102 )
\nSum: ( -4x + 51x = 47x <br/>\ne 35x )—error detected!", "Let’s solve correctly by verifying numerically:
\nTry ( x = 2 ):
\n( 2(2)^2 + 35(2) - 102 = 8 + 70 - 102 = -24 <br/>\ne 0 )—wrong!", "Wait—our factoring was incorrect. Let’s correct the factoring.", "---", "### Correct Factoring Steps", "We need two numbers multiplying to ( 2 \cdot (-102) = -204 ), summing to ( 35 ).", "Try:
\n( 51 \ imes (-4) = -204 ), and ( 51 + (-4) = 47 )
\n( 54 \ imes (-4) = -216 )
\n( 72 \ imes (-3) = -216 )
\n( 68 \ imes (-3) = -204 ), ( 68 - 3 = 65 )
\nEventually, correct pair: 51 and -4 does NOT work.", "After rigorous testing, the correct factorization is:
\n[
\n2x^2 + 35x - 102 = (2x - 6)(x + 17) \quad ? \ ext{ → check: } 2x^2 + 34x - 6x - 102 = 2x^2 + 28x - 102 ❌
\n]", "Alternate approach: use discriminant directly for verification.", "Discriminant:
\n[
\nD = b^2 - 4ac = 35^2 - 4(2)(-102) = 1225 + 816 = 2041
\n]
\nSince ( \sqrt{2041} ) is not a perfect square, roots are irrational:
\n[
\nx = \frac{-35 \pm \sqrt{2041}}{4}
\n]", "But earlier manual trial failed—so double-check by trial and error with possible rational roots from Rational Root Theorem.", "Possible rational roots: factors of ( c/a = -102/2 = -51 ), so ±1, 3, 17, 51, ±½, etc.", "Try ( x = 2 ):
\n( 2(4) + 35(2) - 102 = 8 + 70 - 102 = -24 <br/>\ne 0 )
\nTry ( x = -17 ):
\n( 2(289) + 35(-17) - 102 = 578 - 595 - 102 = -119 <br/>\ne 0 )
\nTry ( x = 3 ):
\n( 2(9) + 35(3) - 102 = 18 + 105 - 102 = 21 <br/>\ne 0 )
\nTry ( x = -12 ):
\n( 2(144) + 35(-12) - 102 = 288 - 420 - 102 = -234 )
\nTry ( x = -2.5 ): ( x = -5/2 )
\n( 2(\frac{25}{4}) + 35(-\frac{5}{2}) - 102 = \frac{50}{4} - \frac{175}{2} - 102 = 12.5 - 87.5 - 102 = -177 ) ❌", "Wait—time to trust the quadratic formula. Despite messy roots, it’s correct.", "So correct roots are:
\n[
\n\boxed{x = \frac{-35 \pm \sqrt{2041}}{4}}
\n]", "Numerically:
\n( \sqrt{2041} \approx 45.18 )
\nSo:
\n( x \approx \frac{-35 + 45.18}{4} = \frac{10.18}{4} \approx 2.545 )
\n( x \approx \frac{-35 - 45.18}{4} = \frac{-80.18}{4} \approx -20.045 )", "Close to earlier mistaken ( x = 2 ) and ( x = -25.5 )? Not close—error in early factoring persists.", "Actually, correct factoring of ( 2x^2 + 35x - 102 ):", "Try grouping:
\n( 2x^2 + 35x - 102 )
\nMultiply ( 2 \cdot (-102) = -204 )
\nFind two numbers ×204, sum 35:
\nAfter testing: 51 and -4 → sum 47
\n48 and -4 → 44
\n52 and -3.92 → no", "Using quadratic formula is definitive:
\n[
\nx = \frac{-35 \pm \sqrt{2041}}{4}
\n]", "---", "### Final Answer", "The solutions to ( 2x^2 + 35x - 102 = 0 ) are:
\n[
\n\boxed{x = \frac{-35 + \sqrt{2041}}{4} \quad \ ext{and} \quad x = \frac{-35 - \sqrt{2041}}{4}}
\n]", "---", "### Why This Matters", "Mastering such equations strengthens your algebra foundation, enables precision in applications like optimization problems, and prepares you for advanced math topics. Use both factoring (when easy) and the quadratic formula (for guarantee) to build confidence.", "---", "### Optimize for SEO: Key Phrases", "- ( solve ( 2x^2 + 35x - 102 = 0 ) )
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\n- algebra techniques for quadratic equations
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\n- irrational roots of ( 2x^2 + 35x - 102 )", "---", "### Conclusion", "Whether you factored correctly or used the quadratic formula, solving ( 2x^2 + 35x - 102 = 0 ) deepens your algebraic intuition. With practice, these equations become familiar and manageable, unlocking broader mathematical insight.", "---", "Keywords: ( 2x^2 + 35x - 102 = 0 ), solve quadratic equation, quadratic formula, factoring quadratic, step-by-step algebra, irrational roots, discriminant calculation, online math solver tips, quadratic solutions.", "---", "Update: Always double-check roots by substitution to validate work!"]
