If you’re wondering about the quadratic equation where the roots are 6 and -4, you’re in the right place. The equation can be easily derived from these roots.
To find the quadratic equation, you can use the fact that if \( r_1 \) and \( r_2 \) are the roots, the equation takes the form \( (x – r_1)(x – r_2) = 0 \).
Thus, the persamaan kuadrat dari yang akar-akarnya 6 dan -4 adalah \( x^2 – 2x – 24 = 0 \). Understanding how to construct this equation opens doors to exploring more about quadratic functions and their properties.
Persamaan Kuadrat Dari Yang Akar-Akarnya 6 dan -4 Adalah
When we talk about quadratic equations, we often encounter terms like roots or solutions. Understanding how to derive a quadratic equation from given roots can be a fun and enlightening journey. In this section, we will dive into how to construct a quadratic equation where the roots are specifically 6 and -4.
Understanding Quadratic Equations
A quadratic equation is typically expressed in the standard form:
\[ ax^2 + bx + c = 0 \]
In this equation:
– **a** represents the coefficient of \( x^2 \) (it’s never zero),
– **b** is the coefficient of \( x \),
– **c** is the constant term.
The solutions to a quadratic equation are known as the roots. If we know the roots, we can easily create our quadratic equation using a specific formula.
Finding the Quadratic Equation from the Roots
If you have the roots of a quadratic equation, you can use the following formula to construct the equation:
If the roots are \( r_1 \) and \( r_2 \), the quadratic equation can be written as:
\[ (x – r_1)(x – r_2) = 0 \]
In our case, the roots \( r_1 \) and \( r_2 \) are 6 and -4. Plugging these values into the formula leads to:
\[ (x – 6)(x + 4) = 0 \]
Let’s expand this expression step by step.
Step 1: Expand the Expression
To expand \( (x – 6)(x + 4) \), we use the distributive property:
– First, multiply \( x \) by both terms in the second bracket:
\[ x \cdot x + x \cdot 4 = x^2 + 4x \]
– Next, multiply \(-6\) by both terms in the second bracket:
\[ -6 \cdot x – 6 \cdot 4 = -6x – 24 \]
Now we combine all the parts:
\[ x^2 + 4x – 6x – 24 \]
Combine like terms:
\[ x^2 – 2x – 24 \]
So, the quadratic equation that has roots 6 and -4 is:
\[ x^2 – 2x – 24 = 0 \]
Exploring the Coefficients
Let’s take a closer look at the coefficients \( a \), \( b \), and \( c \) in our quadratic equation \( x^2 – 2x – 24 = 0 \).
– **Coefficient \( a \)**: In our equation, \( a = 1 \) (the coefficient of \( x^2 \)).
– **Coefficient \( b \)**: Here, \( b = -2 \) (the coefficient of \( x \)).
– **Constant \( c \)**: We find \( c = -24 \).
These coefficients provide insight into the shape and position of the quadratic function when graphed on a coordinate plane.
Graphing the Quadratic Function
Understanding the graph of a quadratic equation helps in visualizing its roots. The graph of \( y = x^2 – 2x – 24 \) is a parabola that opens upwards because the coefficient of \( x^2 \) (which is \( a = 1 \)) is positive.
To find the vertex of the parabola, we can use the vertex formula:
\[ x = -\frac{b}{2a} \]
Substituting our values for \( b \) and \( a \):
\[ x = -\frac{-2}{2 \times 1} = 1 \]
Now, substitute \( x = 1 \) back into the equation to find \( y \):
\[ y = (1)^2 – 2(1) – 24 = 1 – 2 – 24 = -25 \]
So, the vertex of the parabola is at the point (1, -25).
Finding the Y-Intercept
To find the y-intercept of the quadratic function, we need to evaluate the function when \( x = 0 \):
\[ y = (0)^2 – 2(0) – 24 = -24 \]
Thus, the y-intercept is at the point (0, -24).
Identifying the X-Intercepts
The x-intercepts are the roots of the equation, which we found earlier to be:
– \( x = 6 \)
– \( x = -4 \)
On the graph, these points show where the parabola crosses the x-axis.
Characteristics of the Quadratic Equation
Understanding the properties of the quadratic equation \( x^2 – 2x – 24 \) is essential for solving it and applying it in various contexts. Here are some key characteristics:
– **Shape**: The graph is a U-shaped curve (parabola) that opens upwards.
– **Direction**: Since \( a = 1 \) is positive, the parabola opens upwards.
– **Vertex**: The lowest point on the graph is the vertex (1, -25).
– **Y-Intercept**: The graph crosses the y-axis at (0, -24).
– **X-Intercepts**: The graph crosses the x-axis at the roots (6, 0) and (-4, 0).
Applications of Quadratic Equations
Quadratic equations like the one we derived have many applications in real-life scenarios, including:
– **Physics**: Calculating the trajectory of objects under the influence of gravity.
– **Economics**: Modeling profit, cost, and revenue functions.
– **Engineering**: Designing parabolic structures like bridges or satellite dishes.
– **Biology**: Analyzing population growth when conditions change.
Understanding the roots and behavior of quadratic equations helps in various fields and provides valuable insights.
Practice Problems
To solidify your understanding, try solving these practice problems on quadratic equations:
1. Find the quadratic equation that has roots at 3 and -5.
2. What is the vertex of the quadratic equation \( y = 2x^2 – 8x + 6 \)?
3. Determine the x-intercepts of the quadratic equation \( y = x^2 + 4x – 12 \).
4. Analyze the graph of the quadratic equation \( y = -x^2 + 4x + 5 \).
Working through these problems will enhance your grasp of quadratic equations and their properties.
In this journey through the world of quadratic equations, we derived the equation from its roots, explored its graph and properties, and understood its applications. The beauty of mathematics lies in its connections to the real world, and quadratic equations serve as a prime example. Whether you are a student looking to master this topic or simply someone curious about math, understanding quadratic equations enhances your problem-solving skills and analytical thinking.
CARA MENYUSUN PERSAMAAN KUADRAT Jika diketahui Akar-akarnya
Frequently Asked Questions
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What is the standard form of the quadratic equation given the roots 6 and -4?
The standard form of a quadratic equation can be derived from its roots using the formula: \( (x – r_1)(x – r_2) = 0 \), where \( r_1 \) and \( r_2 \) are the roots. Given the roots 6 and -4, the equation becomes \( (x – 6)(x + 4) = 0 \). Expanding this gives \( x^2 – 2x – 24 = 0 \), which is the standard form of the quadratic equation.
How do you find the sum and product of the roots for the quadratic equation?
The sum of the roots of a quadratic equation can be calculated as \( r_1 + r_2 \), while the product is calculated as \( r_1 \times r_2 \). For the roots 6 and -4, the sum is \( 6 + (-4) = 2 \) and the product is \( 6 \times (-4) = -24 \).
Can you explain the relationship between the coefficients and the roots of a quadratic equation?
The coefficients of a quadratic equation \( ax^2 + bx + c = 0 \) relate to the roots through Vieta’s formulas. The sum of the roots \( -\frac{b}{a} \) corresponds to the negative coefficient of \( x \), and the product of the roots \( \frac{c}{a} \) corresponds to the constant term. For our equation \( x^2 – 2x – 24 = 0 \), the coefficient \( b = -2 \) indicates a sum of 2 for the roots, and the constant term \( c = -24 \) indicates a product of -24.
What is the graphical representation of the quadratic equation with roots 6 and -4?
The graphical representation of the quadratic equation \( x^2 – 2x – 24 = 0 \) is a parabola that opens upwards. The x-intercepts of the parabola correspond to the roots at points (6, 0) and (-4, 0). The vertex of the parabola can be found at the midpoint of the roots, which is at \( x = 1 \). Evaluating the quadratic at this point gives the vertex \( (1, -25) \), which is the lowest point of the parabola.
How can one verify the accuracy of the quadratic equation derived from the roots?
To verify the accuracy of the quadratic equation, one should substitute the roots back into the equation. For the derived equation \( x^2 – 2x – 24 = 0 \), substituting \( x = 6 \) results in \( 6^2 – 2(6) – 24 = 0 \) and substituting \( x = -4 \) results in \( (-4)^2 – 2(-4) – 24 = 0 \). Both substitutions should yield true statements, confirming that the roots are indeed correct.
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Final Thoughts
The quadratic equation whose roots are 6 and -4 can be derived using the relationship between roots and coefficients. We use the formula \(x^2 – (r_1 + r_2)x + (r_1 \cdot r_2)\) where \(r_1\) and \(r_2\) are the roots.
In this case, \(r_1 + r_2 = 6 + (-4) = 2\) and \(r_1 \cdot r_2 = 6 \cdot (-4) = -24\).
Thus, the equation becomes \(x^2 – 2x – 24 = 0\). Therefore, the final statement is, ‘persamaan kuadrat dari yang akar-akarnya 6 dan -4 adalah \(x^2 – 2x – 24 = 0\)’.
