If you’re a student or someone who works with numbers, you’ve probably come across quadratic equations at some point. These equations are used to solve problems involving variables raised to the second power, also known as squared variables. One common form of a quadratic equation is 4x^2 – 5x – 12 = 0. In this article, we’ll explore whether this sequence is indeed a quadratic equation and how to solve it using the quadratic formula.
What is a Quadratic Equation?
Before we dive into solving the sequence 4x^2 – 5x – 12 = 0, let’s first understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning it contains a variable raised to the second power. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
In simpler terms, a quadratic equation is an equation that can be written in the form of ax^2 + bx + c = 0, where x represents an unknown value and a, b, and c are known values. The goal is to solve for the value of x that makes the equation true.
Solving Quadratic Equations Using the Quadratic Formula
One of the most common methods for solving quadratic equations is by using the quadratic formula. The quadratic formula is a mathematical formula that provides the solution to any quadratic equation in the form of ax^2 + bx + c = 0. The formula is as follows:
Where a, b, and c are the coefficients of the quadratic equation. The plus-minus symbol indicates that there are two possible solutions to the equation.
Is 4x^2 – 5x – 12 = 0 a Quadratic Equation?
Now that we understand what a quadratic equation is and how to solve it using the quadratic formula, let’s determine if the sequence 4x^2 – 5x – 12 = 0 is indeed a quadratic equation.
Identifying the Coefficients
To determine if the sequence 4x^2 – 5x – 12 = 0 is a quadratic equation, we first need to identify the coefficients of the equation. In this case, a = 4, b = -5, and c = -12.
Verifying the Degree of the Equation
The degree of a polynomial equation is the highest power of the variable in the equation. In a quadratic equation, the degree is always 2. In the sequence 4x^2 – 5x – 12 = 0, the variable x is raised to the second power, making it a quadratic equation.
Conclusion
Based on the above analysis, we can conclude that the sequence 4x^2 – 5x – 12 = 0 is indeed a quadratic equation.
Solving 4x^2 – 5x – 12 = 0 Using the Quadratic Formula
Now that we’ve established that 4x^2 – 5 x – 12 = 0 is a quadratic equation, let’s solve it using the quadratic formula.
Step 1: Identify the Coefficients
As mentioned earlier, the coefficients of the equation are a = 4, b = -5, and c = -12.
Step 2: Substitute the Coefficients into the Quadratic Formula
Substituting the values of a, b, and c into the quadratic formula, we get:
Step 3: Simplify the Equation
Simplifying the equation, we get:
Step 4: Find the Square Root
To solve for x, we need to find the square root of the equation. In this case, the square root of 121 is 11.
Step 5: Solve for x
Finally, we can solve for x by substituting the values of a, b, and c into the simplified quadratic formula:
Solving Quadratic Equations Using the Quadratic Formula Worksheet
To practice solving quadratic equations using the quadratic formula, you can use a worksheet. A quick search online will provide you with various worksheets that you can use to test your skills.
Example Worksheet
The above worksheet provides you with several quadratic equations to solve using the quadratic formula. You can use the steps outlined in this article to solve each equation and check your answers against the provided solutions.
Conclusion
In conclusion, the sequence 4x^2 – 5 x – 12 = 0 is a quadratic equation, and it can be solved using the quadratic formula. Quadratic equations are essential in solving problems involving variables raised to the second power, and the quadratic formula is a useful tool for finding the solutions to these equations. With practice, you can become proficient in solving quadratic equations and use them to solve real-world problems.
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