Find the value of \( x \) if \( 2x^2 - 5x - 3 = 0 \).

["# Find the Value of ( x ) if ( 2x^2 - 5x - 3 = 0 ) — A Complete Guide", "Solving quadratic equations is a fundamental skill in algebra, and understanding how to find the value of ( x ) in equations like ( 2x^2 - 5x - 3 = 0 ) unlocks deeper mathematical insight. Whether you're a student preparing for exams or someone looking to strengthen foundational math skills, knowing how to solve this equation is essential. This article walks you through the process step-by-step, sharing formulas, methods, and practical insights to master quadratic solutions.", "## Understanding the Equation", "The equation ( 2x^2 - 5x - 3 = 0 ) is a standard quadratic equation of the form:", "[\nax^2 + bx + c = 0\n]", "Here, the coefficients are:\n- ( a = 2 )\n- ( b = -5 )\n- ( c = -3 )", "Quadratic equations can be solved using three primary methods: factoring, completing the square, and the quadratic formula. Depending on the equation's form, choosing the right method saves time and ensures accuracy.", "## Method 1: Factoring — The Quickest Approach", "Factoring is often the fastest method when the quadratic can be easily broken down into binomial products. Our goal is to rewrite the equation as:", "[\n(2x + p)(x + q) = 0\n]", "We need two numbers that multiply to ( a \cdot c = 2 \cdot (-3) = -6 ) and add up to ( b = -5 ).", "The pair ( -6 ) and ( +1 ) satisfies both conditions because:\n- ( (-6) \ imes 1 = -6 )\n- ( -6 + 1 = -5 )", "Now, adjust the middle term:", "[\n2x^2 - 6x + x - 3 = 0\n]", "Group and factor by substitution:", "[\n(2x^2 - 6x) + (x - 3) = 0\n\Rightarrow 2x(x - 3) + 1(x - 3) = 0\n\Rightarrow (2x + 1)(x - 3) = 0\n]", "Set each factor equal to zero:\n- ( 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} )\n- ( x - 3 = 0 \Rightarrow x = 3 )", "Solutions from factoring: ( x = -\frac{1}{2} ) or ( x = 3 )", "## Method 2: The Quadratic Formula — A Reliable Fallback", "When factoring is difficult or impossible, the quadratic formula provides a definitive solution:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging in ( a = 2 ), ( b = -5 ), ( c = -3 ):", "[\nx = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}\n= \frac{5 \pm \sqrt{25 + 24}}{4}\n= \frac{5 \pm \sqrt{49}}{4}\n= \frac{5 \pm 7}{4}\n]", "This gives two solutions:\n- ( x = \frac{5 + 7}{4} = \frac{12}{4} = 3 )\n- ( x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2} )", "Verified solutions: ( x = 3 ) and ( x = -\frac{1}{2} ) — matching our factoring results.", "## Why Finding ( x ) Matters", "Finding the value(s) of ( x ) in equations like ( 2x^2 - 5x - 3 = 0 ) introduces key algebraic concepts:\n- Roots and solutions distinguish between real, rational, and irrational solutions.\n- Quadratic equations model real-world scenarios such as projectile motion, profit optimization, and geometry problems.\n- Mastering quadratic formulas and factoring prepares learners for advanced math, including calculus and engineering applications.", "## Step-by-Step Summary", "1. Identify coefficients: ( a = 2 ), ( b = -5 ), ( c = -3 ).\n2. Use factoring or the quadratic formula to solve for ( x ).\n3. Always verify solutions by substituting back into the original equation.\n4. Recognize both roots: ( x = 3 ) and ( x = -\frac{1}{2} ).", "## Final Answer", "The values of ( x ) that satisfy the equation ( 2x^2 - 5x - 3 = 0 ) are:", "[\n\boxed{ x = 3 \quad \ ext{and} \quad x = -\frac{1}{2} }\n]", "Understanding this process empowers you to tackle similar quadratic equations confidently. Practice regularly, and soon solving for ( x ) will feel second nature!"]