Let’s go through a structured approach to problem-solving involving quadratic equations using an example and solving step by step. Quadratic equations are often in the standard form:

$ax^2 + bx + c = 0$

General Problem-Solving Steps:

1. Understand the Problem: Identify the quadratic equation and its context (roots, vertex, maximum/minimum value, etc.).
2. Analyze the Equation: Identify coefficients $a$, $b$, and $c$.
3. Choose a Solving Method: Use one of the following:
   • Factoring (if factorable),
   • Quadratic Formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$),
   • Completing the Square,
   • Graphing.
4. Find the Solution(s): Solve for $x$ or determine other properties as required.
5. Interpret the Result: Analyze the solution in the context of the problem.

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Example Problem:

Solve the quadratic equation $2x^2 - 3x - 2 = 0$.

Step 1: Analyze the coefficients

The given equation is $2x^2 - 3x - 2 = 0$. Identify:

• $a = 2$,
• $b = -3$,
• $c = -2$.

Step 2: Solve using the Quadratic Formula

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

Substitute $a = 2$, $b = -3$, and $c = -2$: $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-2)}}{2(2)}.$

Simplify: $x = \frac{3 \pm \sqrt{9 - (-16)}}{4}.$ $x = \frac{3 \pm \sqrt{9 + 16}}{4}.$ $x = \frac{3 \pm \sqrt{25}}{4}.$ $x = \frac{3 \pm 5}{4}.$

Step 3: Compute the two solutions

1. $x = \frac{3 + 5}{4} = \frac{8}{4} = 2$,
2. $x = \frac{3 - 5}{4} = \frac{-2}{4} = -\frac{1}{2}$.

The solutions are: $x = 2 \quad \text{and} \quad x = -\frac{1}{2}.$

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Step 4: Verify the Solutions

Substitute $x = 2$ and $x = -\frac{1}{2}$ back into the original equation:

1. For $x = 2$: $2(2)^2 - 3(2) - 2 = 8 - 6 - 2 = 0.$ Correct!

2. For $x = -\frac{1}{2}$: $2\left(-\frac{1}{2}\right)^2 - 3\left(-\frac{1}{2}\right) - 2 = 2\left(\frac{1}{4}\right) + \frac{3}{2} - 2 = \frac{1}{2} + \frac{3}{2} - 2 = 0.$ Correct!

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Interpretation:

The roots of the quadratic equation $2x^2 - 3x - 2 = 0$ are: $x = 2 \quad \text{and} \quad x = -\frac{1}{2}.$

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Other Applications of Quadratic Equations:

1. Maximum and Minimum Problems:
   • The vertex $x = -\frac{b}{2a}$ gives the point of maximum or minimum value.
   • Evaluate the vertex for optimization problems (e.g., profit, area).
2. Projectile Motion:
   • Use $y = ax^2 + bx + c$ to calculate the maximum height, range, or time of flight.
3. Graphing:
   • Determine the shape, direction, and key points (vertex, axis of symmetry, and roots).

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Related Questions:

1. What is the geometric interpretation of the roots of a quadratic equation?
2. How can completing the square help rewrite a quadratic equation in vertex form?
3. When does a quadratic equation have no real solutions?
4. What is the discriminant, and how does it determine the number of solutions?
5. How do you apply quadratic equations to solve word problems?

Tip: Always check the discriminant ($\Delta = b^2 - 4ac$) before solving. It helps predict the nature of the roots:

• $\Delta > 0$: Two distinct real roots.
• $\Delta = 0$: One real root (repeated).
• $\Delta < 0$: No real roots (complex roots).