Solve For $x$.$x^2 + 4x - 12 = 0$
**Solve for x: A Step-by-Step Guide to Quadratic Equations** ===========================================================
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. 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.
The Quadratic Equation We're Solving
Our quadratic equation is:
x^2 + 4x - 12 = 0
Why Do We Need to Solve Quadratic Equations?
Quadratic equations are used to model a wide range of real-world problems, such as:
- The trajectory of a projectile
- The motion of an object under the influence of gravity
- The growth of a population
- The design of electrical circuits
How to Solve a Quadratic Equation
There are several methods to solve a quadratic equation, including:
- Factoring
- Quadratic formula
- Graphing
Method 1: Factoring
Factoring involves expressing the quadratic equation as a product of two binomials. In this case, we can factor the equation as:
(x + 6)(x - 2) = 0
Step 1: Identify the Factors
To factor the equation, we need to find two numbers whose product is -12 (the constant term) and whose sum is 4 (the coefficient of the x term). These numbers are 6 and -2.
Step 2: Write the Factored Form
Using the numbers we found in Step 1, we can write the factored form of the equation as:
(x + 6)(x - 2) = 0
Step 3: Solve for x
To solve for x, we set each factor equal to zero and solve for x:
x + 6 = 0 --> x = -6 x - 2 = 0 --> x = 2
Method 2: Quadratic Formula
The quadratic formula is a general method for solving quadratic equations. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Step 1: Identify the Values of a, b, and c
In our equation, a = 1, b = 4, and c = -12.
Step 2: Plug in the Values
Plugging in the values of a, b, and c into the quadratic formula, we get:
x = (-(4) ± √((4)^2 - 4(1)(-12))) / 2(1) x = (-4 ± √(16 + 48)) / 2 x = (-4 ± √64) / 2 x = (-4 ± 8) / 2
Step 3: Solve for x
Simplifying the expression, we get:
x = (-4 + 8) / 2 --> x = 2 x = (-4 - 8) / 2 --> x = -6
Method 3: Graphing
Graphing involves plotting the quadratic equation on a coordinate plane and finding the x-intercepts.
Step 1: Plot the Equation
Plotting the equation x^2 + 4x - 12 = 0 on a coordinate plane, we get a parabola that opens upward.
Step 2: Find the x-Intercepts
The x-intercepts of the parabola are the points where the graph crosses the x-axis. In this case, the x-intercepts are x = -6 and x = 2.
Conclusion
In this article, we solved the quadratic equation x^2 + 4x - 12 = 0 using three different methods: factoring, quadratic formula, and graphing. We found that the solutions to the equation are x = -6 and x = 2.
Frequently Asked Questions
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two.
Q: Why do we need to solve quadratic equations?
A: Quadratic equations are used to model a wide range of real-world problems, such as the trajectory of a projectile, the motion of an object under the influence of gravity, the growth of a population, and the design of electrical circuits.
Q: How do I solve a quadratic equation?
A: There are several methods to solve a quadratic equation, including factoring, quadratic formula, and graphing.
Q: What is the quadratic formula?
A: The quadratic formula is a general method for solving quadratic equations. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation, plug them into the formula, and simplify the expression to find the solutions.
Q: What are the solutions to the quadratic equation x^2 + 4x - 12 = 0?
A: The solutions to the quadratic equation x^2 + 4x - 12 = 0 are x = -6 and x = 2.