Problem 72 - Algebra and Trigonometry

Question

Solve equation using the quadratic formula. $$ 3 x^{2}=6 x-1 $$

Step-by-Step Solution

Answer

The roots of the equation are \(1 - \frac{2\sqrt{3}}{3}\) and \(1 + \frac{2\sqrt{3}}{3}\).

Step 1: Identify a, b, and c

In the given equation, the coefficients in front of \(x^{2}\), \(x\), and the constant term are \(a\), \(b\), and \(c\) respectively. So here, \(a=3\), \(b=-6\), and \(c=-1\).

Step 2: Substitute into the Quadratic Formula

The quadratic formula is \(-b \pm \sqrt{b^{2}-4 a c}\) divided by \(2a\). Substituting \(a=3\), \(b=-6\), and \(c=-1\) into the parentheses gives \(6 \pm \sqrt{(-6)^{2}-4*3*(-1)}\) divided by \(2*3\).

Step 3: Simplify under the square root sign

This gives \(6 \pm \sqrt{36 + 12}\) divided by \(2*3\). This simplifies to \(6 \pm \sqrt{48}\) divided by \(6\).

Step 4: Simplify further

This simplifies to \(6 \pm 4\sqrt{3}\) divided by \(6\). Dividing through by \(6\) specifies the two solutions as follows: \(1 \pm \frac{2\sqrt{3}}{3}\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are fundamental components in algebra and represent an equation of the form \( ax^2 + bx + c = 0 \), where \( a \) is not equal to zero. Solving such equations can seem daunting at first, but there is a powerful and universal tool that can tackle any quadratic equation: the quadratic formula.

This formula is denoted as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) and it provides an efficient way to find the roots of any quadratic equation. Here’s why this method is so valuable: Firstly, it can be applied to any quadratic equation, regardless of the coefficients involved. Secondly, it takes into account every possible scenario, including when the discriminant (\(b^2 - 4ac\)) is positive, negative, or zero, which corresponds to two real solutions, no real solutions, or exactly one real solution, respectively.

Application to an Exercise

Let's apply the quadratic formula to the provided exercise where we are given \(3x^2 = 6x - 1\). The first step is to rearrange the equation into the standard quadratic form to identify the coefficients \(a\), \(b\), and \(c\). This leads us to \(3x^2 - 6x + 1 = 0\) with \(a = 3\), \(b = -6\), and \(c = 1\). Using the quadratic formula, we can substitute these values and solve for \(x\).

The clear, step-by-step approach ensures that students can follow along with the process and see how the formula is applied. This methodology not only aids in solving the given problem but can be applied to any other quadratic equations students may encounter.

Substitution Method in Algebra

The substitution method is another powerful algebraic tool that provides a strategy to solve systems of equations. This method involves isolating one of the variables in one equation and then substituting the expression into the other equation. Doing so reduces the system from two equations in two unknowns to a single equation in one unknown, which is typically easier to solve.

To dig deeper, let's consider the main steps involved in the substitution method:

• First, solve one of the equations for one variable, say \(y\).
• Next, substitute this expression for \(y\) into the other equation.
• Solve the resulting equation for the remaining variable, \(x\).
• Lastly, substitute the value of \(x\) back into the original equation to find the value of \(y\).

Connection to Quadratic Equations

While the substitution method is not directly used in solving the given quadratic equation, understanding how variables can be isolated and substituted is essential for more complex algebra problems and when dealing with systems of equations. The concept of replacing one expression with another equivalent one is fundamental to manipulating and solving algebraic equations.

Simplifying Square Roots

Solving quadratic equations often involves simplifying square roots, as is the case with the quadratic formula's discriminant (\(\sqrt{b^2 - 4ac}\)). Simplifying square roots means expressing the square root in the simplest radical form without a perfect square factoring into the radicand - the number inside the square root.

To simplify square roots, identify any perfect square factors in the radicand. For example, if the radicand is \(48\), you can factor it into \(16 * 3\), where \(16\) is a perfect square. Thus, \(\sqrt{48}\) is simplified to \(\sqrt{16 * 3}\), which can be further broken down to \(4\sqrt{3}\).

In the Context of the Exercise

As in step three of the solution to the given problem, when we calculate the square root of \(48\), we simplify it to \(4\sqrt{3}\) by identifying \(16\) as the perfect square that factors out, making the solution look cleaner and more elegant. Knowing how to simplify square roots is essential for clearly presenting final solutions and for further mathematical manipulation such as adding, subtracting, or multiplying radicals.