Finding the Roots of a Quadratic Equation

The problem asks for the roots of the quadratic equation $6x^2 + 4x - 2 = 0$. The roots of a quadratic equation are the values of the variable $x$ that satisfy the equation. In other words, they are the points where the graph of the equation crosses the x-axis.

We can simplify the given equation by dividing all terms by the common factor, 2:

Now we have a simpler quadratic equation in the standard form $ax^2 + bx + c = 0$, where $a=3$, $b=2$, and $c=-1$. We can find the roots using several methods, such as factorization or the quadratic formula.

Method 1: Factorization

To factor the quadratic $3x^2 + 2x - 1 = 0$, we look for two numbers that multiply to $(a \times c) = (3 \times -1) = -3$ and add up to $b = 2$. These numbers are $3$ and $-1$.

Now, we rewrite the middle term, $2x$, using these two numbers:

Next, we group the terms and factor common factors from each group:

Notice that $(x + 1)$ is a common factor in both terms. We can factor it out:

For this product to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for $x$:

• $x + 1 = 0 \implies x = -1$
• $3x - 1 = 0 \implies 3x = 1 \implies x = \frac{1}{3}$

Thus, the roots of the equation are $x = -1$ and $x = \frac{1}{3}$.

Method 2: Using the Quadratic Formula

The quadratic formula is a general method to find the roots of any quadratic equation $ax^2 + bx + c = 0$. The formula is:

For our simplified equation $3x^2 + 2x - 1 = 0$, we have $a=3$, $b=2$, and $c=-1$. Let's substitute these values into the formula:

This gives us two possible values for $x$:

• $x_1 = \frac{-2 + 4}{6} = \frac{2}{6} = \frac{1}{3}$
• $x_2 = \frac{-2 - 4}{6} = \frac{-6}{6} = -1$

Both methods yield the same roots: $-1$ and $\frac{1}{3}$.

Summary of Roots

The roots of the equation $6x^2 + 4x - 2 = 0$ (or $3x^2 + 2x - 1 = 0$) are $-1$ and $\frac{1}{3}$.

Revision Table: Key Concepts for Roots of Quadratic Equations

Understanding quadratic equations and their roots is fundamental in algebra. Here's a quick review:

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are coefficients and $a \neq 0$.
• Roots: The values of $x$ that satisfy the equation. Also called solutions or zeros.
• Discriminant ($\Delta$): $b^2 - 4ac$. It tells us about the nature of the roots:
  • If $\Delta > 0$: Two distinct real roots.
  • If $\Delta = 0$: One real root (a repeated root).
  • If $\Delta < 0$: Two complex roots.
• Methods to Find Roots:
  • Factorization
  • Completing the Square
  • Quadratic Formula

Additional Information on Solving Quadratic Equations

Quadratic equations appear in many areas of mathematics, physics, and engineering. Being able to find their roots is a crucial skill. While factorization is often the quickest method when applicable, the quadratic formula works for all quadratic equations.

When solving any quadratic equation, it's a good practice to:

1. Write the equation in standard form: $ax^2 + bx + c = 0$.
2. Simplify the equation if possible by dividing by a common factor.
3. Choose a method (factorization, quadratic formula) that seems most appropriate.
4. Carefully perform the calculations.
5. Check your roots by substituting them back into the original equation to ensure they satisfy it. For $x=-1$: $6(-1)^2 + 4(-1) - 2 = 6(1) - 4 - 2 = 6 - 6 = 0$. For $x=1/3$: $6(1/3)^2 + 4(1/3) - 2 = 6(1/9) + 4/3 - 2 = 6/9 + 4/3 - 2 = 2/3 + 4/3 - 2 = 6/3 - 2 = 2 - 2 = 0$. Both roots are correct.

Understanding the relationship between the coefficients ($a, b, c$) and the roots (sum of roots $= -b/a$, product of roots $= c/a$) can also be a useful tool for checking your answers or solving related problems.