Cara Mencari Akar Persamaan Kuadrat

>Hello Sohib EditorOnline, in this article, we will discuss how to find the roots of quadratic equations. Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants and x is the unknown variable. Finding the roots of a quadratic equation is an important skill in mathematics and has many real-world applications. In this article, we will cover different methods of finding the roots of a quadratic equation.

Metode Faktorisasi

The factoring method is the easiest method for finding the roots of a quadratic equation. The basic principle is to factorize the quadratic equation into two linear factors and equate them to zero. Let’s consider the quadratic equation ax² + bx + c = 0. We can write this equation in the form: (px+q)(rx+s) = 0. Expanding this equation, we get: p.r.x² + (p.s+r.q).x + q.s = 0. Equating the coefficients of the variables in both equations, we get:

Coefficients of Equations	Coefficients of Factors
a	p.r
b	p.s + r.q
c	q.s

From the above equations, we can find values of p, q, r, and s by using trial and error method. After finding the values of p, q, r, and s, we can write the quadratic equation in the form of (px+q)(rx+s) = 0. We can then equate each factor to zero and solve for x to get the roots of the quadratic equation.

Contoh Soal

Let’s look at an example to understand the factoring method better. Consider the quadratic equation: 2x² + 7x + 3 = 0. We can factorize this equation into two linear factors as: (2x+1)(x+3) = 0. Equating each factor to zero, we get: 2x+1 = 0 or x+3 = 0. Solving for x, we get the roots of the quadratic equation as: x = -1/2 or x = -3.

Using the factoring method, we can easily find the roots of a quadratic equation if the equation is factorable. However, there are quadratic equations that cannot be factored easily, and in that case, we need to use other methods to find the roots.

Metode Persamaan Kuadrat

The quadratic formula method is a more general method to find the roots of a quadratic equation. The quadratic formula is given by:

x = (-b ± √(b² – 4ac)) / 2a

In this formula, a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0, and ± denotes two roots of the equation. To find the roots of a quadratic equation using the quadratic formula, we need to substitute the values of a, b, and c in the formula and solve for x.

Contoh Soal

Consider the quadratic equation 3x² – 4x – 1 = 0. We can find the roots of this equation using the quadratic formula as follows:

x = (-(-4) ± √((-4)² – 4(3)(-1))) / 2(3)

x = (4 ± √28) / 6

x = (2 ± √7) / 3

So, the roots of the quadratic equation 3x² – 4x – 1 = 0 are (2 + √7) / 3 and (2 – √7) / 3.

Metode Penyelesaian Grafis

The graphical method is another method to find the roots of a quadratic equation. In this method, we plot the quadratic equation on a graph and find the points where the graph intersects the x-axis. The points where the graph intersects the x-axis are the roots of the quadratic equation.

Contoh Soal

Let’s consider the quadratic equation x² – 4x + 4 = 0. We can plot this equation on a graph as shown below:

From the graph, we can see that the graph intersects the x-axis at x = 2. So, the root of the quadratic equation x² – 4x + 4 = 0 is x = 2.

FAQ

Q: Apa itu persamaan kuadrat?

A: Persamaan kuadrat adalah persamaan dengan bentuk ax² + bx + c = 0, di mana a, b, dan c adalah konstanta dan x adalah variabel yang tidak diketahui.

Q: Apa tujuan mencari akar persamaan kuadrat?

A: Mencari akar persamaan kuadrat berguna dalam berbagai aplikasi dunia nyata, seperti dalam ilmu fisika, keuangan, dan ekonomi.

Q: Bagaimana jika persamaan kuadrat tidak bisa difaktorkan?

A: Jika persamaan kuadrat tidak dapat difaktorkan dengan mudah, kita dapat menggunakan metode seperti persamaan kuadrat atau metode numerik lainnya untuk mencari akar persamaan kuadrat.

Q: Apa perbedaan antara metode faktorisasi dan metode persamaan kuadrat?

A: Metode faktorisasi melibatkan faktorisasi persamaan kuadrat menjadi dua faktor linear dan mencari akar dari masing-masing faktor, sedangkan metode persamaan kuadrat melibatkan substitusi koefisien persamaan kuadrat ke dalam rumus persamaan kuadrat dan mencari akar menggunakan rumus tersebut.

Q: Kapan metode penyelesaian grafis digunakan untuk mencari akar persamaan kuadrat?

A: Metode penyelesaian grafis digunakan ketika persamaan kuadrat sulit untuk difaktorkan atau rumus persamaan kuadrat sulit untuk digunakan. Dengan metode ini, kita dapat dengan mudah melihat di mana grafik persamaan kuadrat memotong sumbu-x dan menemukan akarnya.

Cara Mencari Akar Persamaan Kuadrat