When faced with the question, “persamaan kuadrat yang akar-akarnya 5 dan -2 adalah,” the solution reveals itself through simple algebra. To find the quadratic equation with roots at 5 and -2, we can use the fact that the equation can be formed as \( (x – 5)(x + 2) = 0 \).
Multiplying this out gives us \( x^2 – 3x – 10 = 0 \). This equation not only encapsulates these specific roots but also opens the door to understanding the fundamentals of quadratic equations. Let’s delve deeper into how we arrive at this expression and explore its implications in algebra.
Persamaan Kuadrat yang Akar-Akarnya 5 dan -2 Adalah
In mathematics, quadratic equations play an essential role. They appear in various fields like physics, engineering, and economics. The specific quadratic equation that has roots (or solutions) of 5 and -2 is particularly interesting. In this article, we will delve deep into understanding what a quadratic equation is and how one can derive it from its roots. We will also explore related concepts, applications, and examples to provide a comprehensive understanding of the topic.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, which means it involves the square of a variable. The standard form of a quadratic equation is given by:
\[ ax^2 + bx + c = 0 \]
Where:
– \( a, b, c \) are constants, and \( a \neq 0 \).
– \( x \) represents the variable.
When we say that the roots of a quadratic equation are 5 and -2, we are referring to the solutions of this equation. In other words, if we substitute \( x = 5 \) or \( x = -2 \) into the equation, it should equal zero.
Finding the Quadratic Equation from Its Roots
If we know the roots of a quadratic equation, we can find the equation using the factored form:
\[ (x – r_1)(x – r_2) = 0 \]
Where \( r_1 \) and \( r_2 \) are the roots. In our case, the roots \( r_1 = 5 \) and \( r_2 = -2 \). Thus, we can write:
\[ (x – 5)(x + 2) = 0 \]
Now, let’s expand this expression:
- First, multiply \( x \) by \( x \):
\( x^2 \) - Next, multiply \( x \) by \( 2 \):
\( +2x \) - Then, multiply \( -5 \) by \( x \):
\( -5x \) - Finally, multiply \( -5 \) by \( 2 \):
\( -10 \)
Combining these results, we have:
\[ x^2 + 2x – 5x – 10 = 0 \]
This simplifies to:
\[ x^2 – 3x – 10 = 0 \]
So, the quadratic equation whose roots are 5 and -2 is:
\[ x^2 – 3x – 10 = 0 \]
Verifying the Roots
To ensure that our quadratic equation is correct, we can verify that substituting the roots back into the equation results in zero.
1. **For root 5:**
\[
5^2 – 3(5) – 10 = 25 – 15 – 10 = 0
\]
2. **For root -2:**
\[
(-2)^2 – 3(-2) – 10 = 4 + 6 – 10 = 0
\]
Both calculations confirm that 5 and -2 are indeed the roots of the equation \( x^2 – 3x – 10 = 0 \).
The Importance of Quadratic Equations
Quadratic equations and their solutions have practical significance in real life:
– **Physics**: They model projectile motion, determining the path of thrown objects.
– **Engineering**: They are used in design and optimization problems.
– **Economics**: They analyze profit maximization and cost minimization.
Understanding how to derive and solve them is crucial for students and professionals in these fields.
Solving Quadratic Equations
There are several methods to solve quadratic equations. Here, we will discuss the three most common methods.
1. **Factoring**:
This is the method we used above, where we factor the quadratic into simpler binomials and find the roots.
2. **Using the Quadratic Formula**:
The quadratic formula provides a straightforward way to find the roots for any quadratic equation:
\[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
\]
For our equation \( x^2 – 3x – 10 = 0 \):
– \( a = 1 \)
– \( b = -3 \)
– \( c = -10 \)
Plugging these values into the formula gives:
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 – 4(1)(-10)}}{2(1)}
\]
\[
x = \frac{3 \pm \sqrt{9 + 40}}{2} = \frac{3 \pm \sqrt{49}}{2} = \frac{3 \pm 7}{2}
\]
This results in:
\[
x = \frac{10}{2} = 5 \quad \text{and} \quad x = \frac{-4}{2} = -2
\]
Thus, we again arrive at the roots of 5 and -2.
3. **Completing the Square**:
This method involves rearranging the equation to form a perfect square. It can be a bit complex but is extremely useful.
Starting from our equation \( x^2 – 3x – 10 = 0 \):
– Rearrange to isolate the constant:
\[
x^2 – 3x = 10
\]
– To complete the square, take half of the coefficient of \( x \) (which is -3), square it, and add it to both sides:
\[
(-\frac{3}{2})^2 = \frac{9}{4}
\]
– Add \( \frac{9}{4} \) to both sides:
\[
x^2 – 3x + \frac{9}{4} = 10 + \frac{9}{4}
\]
\[
x^2 – 3x + \frac{9}{4} = \frac{40}{4} + \frac{9}{4} = \frac{49}{4}
\]
– This can be factored into:
\[
(x – \frac{3}{2})^2 = \frac{49}{4}
\]
– Taking the square root of both sides gives us:
\[
x – \frac{3}{2} = \pm \frac{7}{2}
\]
– Solving these equations leads to the same roots of \( x = 5 \) and \( x = -2 \).
Graphing Quadratic Equations
Graphing quadratic equations helps visualize their properties. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \).
For our quadratic equation \( y = x^2 – 3x – 10 \):
– The vertex can be found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{-3}{2 \cdot 1} = \frac{3}{2}
\]
– To find the y-coordinate of the vertex, substitute \( x = \frac{3}{2} \) back into the equation:
\[
y = \left(\frac{3}{2}\right)^2 – 3\left(\frac{3}{2}\right) – 10
\]
\[
= \frac{9}{4} – \frac{9}{2} – 10
\]
\[
= \frac{9}{4} – \frac{18}{4} – \frac{40}{4} = -\frac{49}{4}
\]
Thus, the vertex is at \( \left(\frac{3}{2}, -\frac{49}{4}\right) \).
To graph the quadratic:
– Start by plotting the vertex.
– Mark the roots (5 and -2) on the x-axis.
– Draw a smooth curve through these points that opens upwards since the coefficient of \( x^2 \) is positive.
Applications of Quadratic Equations in Real Life
Quadratic equations are not just theoretical; they have numerous applications in many fields. Let’s explore a few practical examples:
1. **Projectile Motion**: When you throw a ball, its path resembles a parabola. You can use quadratic equations to predict its height at various points.
2. **Area Problems**: Suppose you have a rectangular garden. If the length of the garden is expressed in terms of its width using a quadratic equation, you can determine the dimensions that maximize the area.
3. **Economics**: Businesses often use quadratic equations to identify the best pricing strategy to maximize profits.
4. **Engineering**: Engineers apply quadratic equations when designing structures—ensuring they can withstand forces encountered throughout their lifespan.
Conclusion
Understanding quadratic equations, particularly those with specific roots like 5 and -2, provides valuable insights into mathematics and its applications. The ability to derive these equations, solve them by various methods, and understand their real-world implications is
Persamaan kuadrat yang akar-akarnya 5 dan -6 adalah ….
Frequently Asked Questions
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How can I derive the quadratic equation from the roots 5 and -2?
To derive the quadratic equation from the roots 5 and -2, you can use the fact that if r1 and r2 are the roots, the quadratic can be expressed as (x – r1)(x – r2) = 0. Substituting the roots, you get (x – 5)(x + 2) = 0. Expanding this expression results in x² – 3x – 10 = 0. Thus, the quadratic equation is x² – 3x – 10 = 0.
What are the steps to solve the quadratic equation x² – 3x – 10 = 0?
You can solve the quadratic equation x² – 3x – 10 = 0 by factoring, completing the square, or using the quadratic formula. Factoring gives you (x – 5)(x + 2) = 0, leading to the solutions x = 5 and x = -2. Alternatively, using the quadratic formula x = [-b ± √(b² – 4ac)] / (2a) also yields the same roots when applying a = 1, b = -3, and c = -10.
What is the significance of the coefficients in the quadratic equation x² – 3x – 10 = 0?
The coefficients in the quadratic equation x² – 3x – 10 = 0 provide important information about the roots and the graph of the equation. The coefficient of x² (which is 1) indicates that the parabola opens upwards. The coefficient of x (which is -3) affects the position of the vertex, while the constant term (-10) determines where the parabola crosses the y-axis. Together, these coefficients help define the shape and position of the graph.
Can you explain how to find the vertex of the quadratic equation x² – 3x – 10 = 0?
To find the vertex of the quadratic equation x² – 3x – 10 = 0, use the formula for the x-coordinate of the vertex, which is x = -b/(2a). Here, a is 1 and b is -3, so x = -(-3)/(2 * 1) = 3/2. To find the y-coordinate, substitute x = 3/2 back into the equation: y = (3/2)² – 3(3/2) – 10. This calculation gives the vertex point as (3/2, – 49/4).
What graphical features can I expect from the parabola represented by the equation x² – 3x – 10 = 0?
The parabola represented by the equation x² – 3x – 10 = 0 opens upwards due to the positive coefficient of x². Its vertex is located at (3/2, -49/4), which serves as the minimum point of the graph. The roots at x = 5 and x = -2 indicate that the graph intersects the x-axis at those points. Additionally, the y-intercept occurs at (0, -10), where the graph intersects the y-axis.
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Final Thoughts
Persamaan kuadrat yang akar-akarnya 5 dan -2 adalah \(x^2 – 3x – 10 = 0\). Kita mendapatkan persamaan ini dengan menggunakan rumus akar kuadrat. Dalam hal ini, \(p = 5 + (-2) = 3\) dan \(q = 5 \times -2 = -10\).
Dengan menyusun kembali, kita bisa membentuk persamaan kuadrat yang tepat. Persamaan ini memiliki dua akar, yaitu 5 dan -2, seperti yang kita inginkan. Dengan memahami cara menyusun persamaan ini, kita bisa lebih mudah mengerjakan masalah matematika yang serupa.
