#### \( 2x^2 + 4x - 30 = 0 \)

["# Solving the Quadratic Equation ( 2x^2 + 4x - 30 = 0 ): A Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and math enthusiasts alike. One common equation encountered is:", "[\n2x^2 + 4x - 30 = 0\n]", "Whether you're tackling this problem for school, algebra practice, or simply to strengthen your math abilities, this article will walk you through solving the equation step-by-step, explain how to find the roots, and cover related concepts like vertex form, discriminant, and graphing.", "---", "## What Is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation of the form:", "[\nax^2 + bx + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants and ( a <br/>\ne 0 ). The solutions or roots represent the x-values where the quadratic function intersects the x-axis.", "---", "## Step 1: Simplify the Equation (If Possible)", "Start with the original equation:", "[\n2x^2 + 4x - 30 = 0\n]", "Check if all terms can be divided by a common factor. Here, each coefficient is divisible by 2. Divide the entire equation by 2:", "[\nx^2 + 2x - 15 = 0\n]", "This simplified version is easier to solve using factoring, the quadratic formula, or completing the square.", "---", "## Step 2: Solve for ( x ) Using the Quadratic Formula", "The most reliable method for solving any quadratic equation is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For the simplified equation ( x^2 + 2x - 15 = 0 ):", "- ( a = 1 )\n- ( b = 2 )\n- ( c = -15 )", "Plug in these values:", "[\nx = \frac{-(2) \pm \sqrt{(2)^2 - 4(1)(-15)}}{2(1)}\n]", "[\nx = \frac{-2 \pm \sqrt{4 + 60}}{2}\n]", "[\nx = \frac{-2 \pm \sqrt{64}}{2}\n]", "[\nx = \frac{-2 \pm 8}{2}\n]", "Now calculate both solutions:", "1. ( x = \frac{-2 + 8}{2} = \frac{6}{2} = 3 )", "2. ( x = \frac{-2 - 8}{2} = \frac{-10}{2} = -5 )", "---", "## Final Roots", "The solutions (roots) of the equation ( 2x^2 + 4x - 30 = 0 ) are:", "[\n\boxed{x = 3} \quad \ ext{and} \quad \boxed{x = -5}\n]", "These mean the quadratic intersects the x-axis at ( x = 3 ) and ( x = -5 ).", "---", "## Step 3: Verify Solutions by Substitution", "To confirm the roots are correct, substitute ( x = 3 ) and ( x = -5 ) back into the original equation.", "For ( x = 3 ):", "[\n2(3)^2 + 4(3) - 30 = 2(9) + 12 - 30 = 18 + 12 - 30 = 0 \quad \ ext{✓}\n]", "For ( x = -5 ):", "[\n2(-5)^2 + 4(-5) - 30 = 2(25) - 20 - 30 = 50 - 20 - 30 = 0 \quad \ ext{✓}\n]", "Both solutions satisfy the equation.", "---", "## Prove the Roots via Factoring (Optional)", "Since we used the quadratic formula, let’s explore factoring for insight:", "From earlier, the simplified equation is ( x^2 + 2x - 15 = 0 ).", "Find two numbers that multiply to ( -15 ) and add to ( 2 ): ( 5 ) and ( -3 ).", "So,", "[\n(x + 5)(x - 3) = 0\n]", "Setting each factor to zero:", "[\nx + 5 = 0 \Rightarrow x = -5\n]\n[\nx - 3 = 0 \Rightarrow x = 3\n]", "Confirms our earlier results.", "---", "## Visualize the Quadratic Function", "The graph of the equation ( y = 2x^2 + 4x - 30 ) is a parabola opening upward because the coefficient of ( x^2 ) is positive.", "Since we found the roots at ( x = -5 ) and ( x = 3 ), the parabola crosses the x-axis at these points. The vertex lies halfway between the roots, helping determine the minimum point.", "The vertex ( x )-coordinate is:", "[\nx = \frac{-b}{2a} = \frac{-4}{2(2)} = -1\n]", "Plug in ( x = -1 ) to find the y-coordinate:", "[\ny = 2(-1)^2 + 4(-1) - 30 = 2 - 4 - 30 = -32\n]", "Vertex: ( (-1, -32) )", "---", "## Related Concepts & Forms", "### 1. Vertex Form", "Transform the equation into vertex form ( y = a(x - h)^2 + k ):", "[\n\begin{aligned}\ny &= 2x^2 + 4x - 30 \\n&= 2(x^2 + 2x) - 30 \\n&= 2(x^2 + 2x + 1 - 1) - 30 \\n&= 2((x + 1)^2 - 1) - 30 \\n&= 2(x + 1)^2 - 2 - 30 \\n&= 2(x + 1)^2 - 32\n\end{aligned}\n]", "Vertex: ( (-1, -32) )", "---", "### 2. Discriminant Insight", "The discriminant ( D = b^2 - 4ac ) tells us about the nature of the roots:", "[\nD = (4)^2 - 4(2)(-30) = 16 + 240 = 256\n]", "Since ( D > 0 ), there are two distinct real roots — consistent with our solutions.", "---", "## Why This Equation Matters", "Quadratic equations like ( 2x^2 + 4x - 30 = 0 ) appear in many real-world scenarios, including:", "- Modeling projectile motion (e.g., when will a ball hit the ground)\n- Calculating profit or loss in economics\n- Engineering problems involving parabolic paths", "Understanding how to solve and interpret these equations is valuable for students and professionals in STEM fields.", "---", "## Summary", "The quadratic equation:", "[\n2x^2 + 4x - 30 = 0\n]", "has two real solutions:", "[\n\boxed{x = 3} \quad \ ext{and} \quad \boxed{x = -5}\n]", "These roots can be verified algebraically, graphically, or via factoring. Learning to solve such equations builds a strong foundation in algebra and problem-solving.", "---", "## Need More Help with Quadratics?", "- Try solving other quadratics using the quadratic formula\n- Explore graphing tools to visualize parabolas\n- Understand applications in physics, economics, and geometry", "Mastering quadratics empowers you to tackle complex real-world challenges — start mastering them today!", "---", "Keywords: solve ( 2x^2 + 4x - 30 = 0 ), quadratic equation solutions, quadratic formula, factoring quadratics, vertex form, discriminant, algebra practice, quadratic roots, graph quadratic equation."]