\[ 2u^2 + 3u - 2 = 0 \] - United Radiology

["Solving the Quadratic Equation ( 2u^2 + 3u - 2 = 0 ): Step-by-Step Guide with Methods and Solutions", "Solving quadratic equations is a fundamental skill in algebra, essential for students, academics, and professionals alike. One of the most commonly encountered quadratics is the equation:", "[
\n2u^2 + 3u - 2 = 0
\n]", "In this article, we’ll explore multiple methods to solve this equation, understand the steps clearly, and interpret the real-world applications of its solutions. Whether you're preparing for exams, tackling math problems, or simply deepening your algebra knowledge, this guide will walk you through finding the roots of ( 2u^2 + 3u - 2 = 0 ) with precision and confidence.", "---", "### What is a Quadratic Equation?", "A quadratic equation has the standard form:", "[
\nau^2 + bu + c = 0
\n]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\ne 0 ). In our case, ( a = 2 ), ( b = 3 ), and ( c = -2 ). These equations form a parabola when graphed, and their solutions—called roots—represent the ( u )-values where the graph intersects the ( u )-axis.", "---", "### How to Solve ( 2u^2 + 3u - 2 = 0 ): Methods Explained", "There are several reliable methods to solve this quadratic equation. We’ll cover elimination, completing the square, and the quadratic formula—each providing unique insight into the nature of the roots.", "---", "#### Method 1: Quadratic Formula", "The quadratic formula is the most universal method and works for any quadratic equation:", "[
\nu = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Plug in ( a = 2 ), ( b = 3 ), ( c = -2 ):", "[
\nu = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm \sqrt{25}}{4}
\n]", "Since ( \sqrt{25} = 5 ):", "[
\nu = \frac{-3 \pm 5}{4}
\n]", "So, two solutions emerge:", "[
\nu = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}
\n]", "[
\nu = \frac{-3 - 5}{4} = \frac{-8}{4} = -2
\n]", "Solutions:
\n[
\nu = \frac{1}{2} \quad \ ext{and} \quad u = -2
\n]", "---", "#### Method 2: Factoring", "Factoring works well when the quadratic factors neatly. Start by multiplying ( a \cdot c = 2 \cdot (-2) = -4 ). We look for two numbers that multiply to (-4) and add up to ( b = 3 ). Those numbers are ( 4 ) and (-1).", "Split the middle term:", "[
\n2u^2 + 4u - u - 2 = 0
\n]", "Group:", "[
\n(2u^2 + 4u) + (-u - 2) = 0
\n]
\n[
\n2u(u + 2) -1(u + 2) = 0
\n]
\n[
\n(2u - 1)(u + 2) = 0
\n]", "Set each factor equal to zero:", "[
\n2u - 1 = 0 \Rightarrow u = \frac{1}{2}, \quad u + 2 = 0 \Rightarrow u = -2
\n]", "Same solutions confirmed:
\n[
\n\boxed{u = \frac{1}{2}} \quad \ ext{and} \quad \boxed{u = -2}
\n]", "---", "#### Method 3: Completing the Square", "This method demonstrates algebraic manipulation to isolate ( u ).", "Start with:", "[
\n2u^2 + 3u - 2 = 0
\n]", "Move constant term to the other side:", "[
\n2u^2 + 3u = 2
\n]", "Divide entire equation by 2 to simplify the coefficient of ( u^2 ):", "[
\nu^2 + \frac{3}{2}u = 1
\n]", "To complete the square, take half of ( \frac{3}{2} ), which is ( \frac{3}{4} ), and square it: ( \left( \frac{3}{4} \right)^2 = \frac{9}{16} ). Add this to both sides:", "[
\nu^2 + \frac{3}{2}u + \frac{9}{16} = 1 + \frac{9}{16}
\n]
\n[
\n\left( u + \frac{3}{4} \right)^2 = \frac{16}{16} + \frac{9}{16} = \frac{25}{16}
\n]", "Take the square root of both sides:", "[
\nu + \frac{3}{4} = \pm \frac{5}{4}
\n]", "Solve for ( u ):", "[
\nu = -\frac{3}{4} \pm \frac{5}{4}
\n]", "So,", "[
\nu = -\frac{3}{4} + \frac{5}{4} = \frac{2}{4} = \frac{1}{2}, \quad u = -\frac{3}{4} - \frac{5}{4} = -\frac{8}{4} = -2
\n]", "Again, we arrive at:
\n[
\n\boxed{u = \frac{1}{2}} \quad \ ext{and} \quad \boxed{u = -2}
\n]", "---", "### What Do These Solutions Mean?", "The equation ( 2u^2 + 3u - 2 = 0 ) models real-world relationships in physics, economics, and engineering. The roots represent:", "- Critical points, such as break-even values in cost-profit analysis.
\n- Times or positions in motion equations, where the object hits a determined measurement (e.g., position = 0).
\n- Inputs that yield a specific output in mathematical models.", "For instance, if ( u ) represents time and the equation models displacement, ( u = -2 ) and ( u = \frac{1}{2} ) could indicate past or future moments when a certain position is reached.", "---", "### Verification by Substitution", "Always verify solutions by plugging back into the original equation.", "For ( u = \frac{1}{2} ):", "[
\n2\left(\frac{1}{2}\right)^2 + 3\left(\frac{1}{2}\right) - 2 = 2 \cdot \frac{1}{4} + \frac{3}{2} - 2 = \frac{1}{2} + 1.5 - 2 = 0
\n]", "For ( u = -2 ):", "[
\n2(-2)^2 + 3(-2) - 2 = 8 - 6 - 2 = 0
\n]", "Both satisfy the equation—confirming our solutions.", "---", "### Summary", "The quadratic equation ( 2u^2 + 3u - 2 = 0 ):", "- Has two real solutions: ( u = \frac{1}{2} ) and ( u = -2 ).
\n- Can be solved using multiple methods: quadratic formula, factoring, and completing the square.
\n- Has practical significance across scientific and industrial fields.
\n- Requires careful step-by-step execution and verification to ensure accuracy.", "---", "### Frequently Asked Questions (FAQs)", "Q: Can quadratic equations have complex roots?
\nA: Yes, if the discriminant (( b^2 - 4ac )) is negative, roots are complex conjugates. In this equation, the discriminant is positive, so roots are real and distinct.", "Q: Why factoring works only sometimes?
\nA: Factoring depends on the ability to split the middle term into integers or simple fractions. Not all quadratics factor nicely, which is why the formula and completing the square are universal methods.", "Q: How can I graph this equation?
\nA: Plotting ( y = 2u^2 + 3u - 2 ) shows a parabola opening upward (since ( a > 0 )), crossing the ( u )-axis at ( u = -2 ) and ( u = \frac{1}{2} ).", "---", "### Final Thoughts", "Mastering quadratics like ( 2u^2 + 3u - 2 = 0 ) opens doors to deeper algebraic understanding and real-world problem solving. Whether you’re a student, teacher, or self-learner, knowing multiple solution methods builds flexibility and confidence. Use this guide as a foundation—practice more equations, explore advanced topics like discriminant analysis, and keep building your mathematical fluency!", "---", "Keywords: quadratic equation, solve ( 2u^2 + 3u - 2 = 0 ), quadratic formula, factoring, completing the square, algebra 2, real solutions, discriminant, verification steps, math solutions"]