What is a Quadratic Equation and How to Solve It?
Quick Answer
A quadratic equation is a polynomial equation of degree two, typically in the form ax² + bx + c = 0. To solve it, you can use substitution or the quadratic formula, ensuring to simplify the equation first.
A quadratic equation is a second-degree polynomial equation that can be expressed in the standard form: ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The highest exponent of the variable 'x' is 2, which gives the equation its name. These equations are prevalent in various fields such as physics, engineering, and finance, where they can model trajectories, optimize outcomes, and represent relationships between variables.
To solve a quadratic equation, you can use several methods, including factoring, completing the square, or applying the quadratic formula. The quadratic formula is particularly useful because it can provide solutions for any quadratic equation, regardless of whether it can be factored easily. The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
This formula allows you to find the values of 'x' that satisfy the equation. Let's look at an example to see how this works in practice.
Consider the equation 3x² - 4x - 4 = 3. The first step to solving this equation is to rearrange it into standard form. We start by moving all terms to one side by subtracting 3 from both sides:
3x² - 4x - 4 - 3 = 0
=> 3x² - 4x - 7 = 0.
Now we can test potential solutions for 'x'. In this case, we want to check if x = -1 and x = 7/3 are solutions of the equation. To do this, we substitute these values back into the equation:
1. For x = -1:
3(-1)² - 4(-1) - 7 = 3(1) + 4 - 7 = 3 + 4 - 7 = 0, which confirms that x = -1 is a solution.
2. For x = 7/3:
3(7/3)² - 4(7/3) - 7 = 3 * (49/9) - (28/3) - 7 = (49/3) - (28/3) - (21/3) = (49 - 28 - 21)/3 = 0, confirming that x = 7/3 is also a solution.
Both x = -1 and x = 7/3 satisfy the original equation, making them valid solutions. Quadratic equations can be graphed as parabolas, and the solutions correspond to the points where the parabola intersects the x-axis. Understanding how to solve quadratic equations is crucial for higher-level math and real-world problem solving. Practice with different equations will enhance your skills and confidence in tackling quadratic equations.
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