The solution to 3×2 12x 24 0 is – Embark on a mathematical journey to solve the enigmatic equation 3x² + 12x + 24 = 0. Our adventure begins with a deep dive into the world of algebraic equations, uncovering their secrets and unlocking the path to their solutions.
We will explore the intricacies of linear equations, mastering the art of solving them with precision. Then, we delve into the realm of trinomials, learning how to factor them into simpler forms. The quadratic formula will become our trusty companion, guiding us through the complexities of quadratic equations.
Algebraic Equation
An algebraic equation is a mathematical statement that shows the equality of two expressions, typically containing variables and constants. It’s a fundamental concept in algebra, forming the basis for solving problems and modeling real-world scenarios.
An algebraic equation consists of:
- Variables:Unknown values represented by letters (e.g., x, y, z).
- Constants:Known values represented by numbers or letters (e.g., 5, a).
- Operators:Symbols that indicate mathematical operations (e.g., +, -, -, /).
- Equality sign (=):Indicates that the expressions on both sides are equal.
Example, The solution to 3×2 12x 24 0 is
Consider the equation 2x + 5 = 13. Here, ‘x’ is the variable, ‘2’ and ‘5’ are constants, ‘+’ is the addition operator, and ‘=’ is the equality sign. To solve this equation, we isolate the variable on one side by performing inverse operations (adding or subtracting the same value from both sides) until we find the value of ‘x’.
Solving Linear Equations: The Solution To 3×2 12x 24 0 Is
Solving a linear equation involves finding the value of a variable that makes the equation true. It is a fundamental operation in algebra, and it is used in a wide variety of applications, including problem-solving, data analysis, and modeling.
Steps Involved in Solving a Linear Equation
- Simplify the equation:Remove any parentheses or fractions and combine like terms.
- Isolate the variable term:Move all terms with the variable to one side of the equation and all constants to the other side.
- Solve for the variable:Divide both sides of the equation by the coefficient of the variable to isolate the variable.
Example, The solution to 3×2 12x 24 0 is
Let’s solve the linear equation 2x + 5 = 13:
- Simplify:2x + 5 = 13
- Isolate the variable term:2x = 8
- Solve for the variable:x = 4
Therefore, the solution to the equation 2x + 5 = 13 is x = 4.
Table of Steps
| Step | Description |
|---|---|
| 1 | Simplify the equation |
| 2 | Isolate the variable term |
| 3 | Solve for the variable |
Factoring Trinomials
Factoring trinomials is a technique for expressing a trinomial (an expression with three terms) as a product of two binomials (expressions with two terms). This process involves finding two numbers that, when multiplied, give the constant term and when added, give the coefficient of the middle term.
Methods for Factoring Trinomials
- Trinomials with a Leading Coefficient of 1:For trinomials in the form x² + bx + c, find two numbers whose product is c and whose sum is b. For example, to factor x² + 5x + 6, we find that 2 and 3 have a product of 6 and a sum of 5, so x² + 5x + 6 = (x + 2)(x + 3).
- Trinomials with a Leading Coefficient Other Than 1:For trinomials in the form ax² + bx + c, where a ≠ 1, first factor out the greatest common factor (GCF) from the first two terms. Then, factor the remaining trinomial using the methods for trinomials with a leading coefficient of 1.
- Trinomials with a Negative Leading Coefficient:For trinomials in the form -ax² + bx + c, where a ≠ 0, first factor out -a from the entire trinomial. Then, factor the remaining trinomial using the methods for trinomials with a leading coefficient of 1.
Table of Factoring Methods
| Trinomial Type | Factoring Method | Example ||—|—|—|| x² + bx + c | Find two numbers whose product is c and whose sum is b | x² + 5x + 6 = (x + 2)(x + 3) || ax² + bx + c | Factor out the GCF, then factor the remaining trinomial | 2x² + 8x + 6 = 2(x² + 4x + 3) = 2(x + 3)(x + 1) ||
- ax² + bx + c | Factor out
- a, then factor the remaining trinomial |
- x² + 5x + 6 =
- (x²
- 5x
- 6) =
- (x
- 6)(x + 1) |
Quadratic Formula
The quadratic formula is a mathematical equation used to solve quadratic equations of the form ax2+ bx+ c= 0, where a, b, and care real numbers and ais not equal to 0.
Significance of the Quadratic Formula
The quadratic formula provides a general method for solving quadratic equations, regardless of their complexity or the values of the coefficients a, b, and c. It eliminates the need for using specific techniques or trial-and-error methods to find the roots of quadratic equations.
How to Use the Quadratic Formula
To solve a quadratic equation using the quadratic formula, follow these steps:
Identify the coefficients
Determine the values of a, b, and cin the quadratic equation ax2+ bx+ c= 0.
Substitute into the formula
Replace a, b, and cin the quadratic formula x= (- b± √( b24 ac)) / 2 a.
-
Calculate the discriminant
Determine the value of the discriminant, b2
- 4 ac.
Solve for x
Calculate the two possible values of xby plugging the values of a, b, and cinto the quadratic formula.
Completing the Square
Completing the square is a technique used to solve quadratic equations. It involves adding and subtracting a constant term to the equation to make it a perfect square trinomial, which can then be easily factored and solved.
Example, The solution to 3×2 12x 24 0 is
Let’s consider the quadratic equation x2+ 6 x– 7 = 0. To complete the square, we need to add and subtract the square of half the coefficient of the x-term, which is 6/2 = 3. Thus, we have:
“`x 2+ 6 x
- 7 + 3 2
- 3 2= 0
x 2+ 6 x+ 9
- 7
- 9 = 0
( x+ 3) 2
16 = 0
“`
Now, we can solve for xby isolating the perfect square term:
“`( x+ 3) 2= 16 x+ 3 = ±4 x=
3 ± 4
x=
7 or x= 1
“`
Comparison with Quadratic Formula
Completing the square and using the quadratic formula are two methods for solving quadratic equations. Here’s a table comparing the two methods:
| Method | Formula | Advantages | Disadvantages |
|---|---|---|---|
| Completing the Square | x =
|
– Works well when the coefficient of x2 is 1
|
– Can be more complex for equations with large coefficients |
| Quadratic Formula | x= (- b± √( b2
|
– Works for all quadratic equations
|
– Can be cumbersome for equations with small coefficients |
FAQ Section
What is an algebraic equation?
An algebraic equation is a mathematical statement that uses variables to represent unknown values.
How do you solve a linear equation?
To solve a linear equation, isolate the variable term on one side of the equation and the constant term on the other side.
What is the quadratic formula?
The quadratic formula is a mathematical formula that can be used to solve quadratic equations of the form ax² + bx + c = 0.
