Problem 76 - Elementary Algebra

Question

Solve. $$ 3 x 2-13 x+4=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = \frac{1}{3}\).

Step 1: Rearrange Equation

First, rewrite the equation in standard quadratic form, which should look like \(ax^2 + bx + c = 0\). Our equation given is improperly formatted, so rearrange as \(3x^2 - 13x + 4 = 0\).

Step 2: Identify Coefficients

Identify the coefficients from the quadratic equation \(ax^2 + bx + c = 0\). In this equation, \(a = 3\), \(b = -13\), and \(c = 4\).

Step 3: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Substituting our coefficients in gives us \(x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4 \times 3 \times 4}}{2 \times 3}\).

Step 4: Calculate the Discriminant

Calculate \(b^2 - 4ac\), which is the discriminant: \((-13)^2 - 4 \times 3 \times 4 = 169 - 48 = 121\).

Step 5: Simplify the Formula

Substitute the discriminant back into the quadratic formula: \(x = \frac{13 \pm \sqrt{121}}{6}\). Since \(\sqrt{121} = 11\), simplify further: \(x = \frac{13 \pm 11}{6}\).

Step 6: Solve for x

Solve for the two possible solutions by splitting into two equations: \(x = \frac{13 + 11}{6} = 4\) and \(x = \frac{13 - 11}{6} = \frac{1}{3}\).

Key Concepts

Quadratic Formula

The quadratic formula is a tool that allows us to find the solutions of a quadratic equation. This remarkable formula is written as:

• \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

It uses the coefficients \(a\), \(b\), and \(c\) from the standard form of a quadratic equation \(ax^2 + bx + c = 0\). By plugging these values into the quadratic formula, we can find the values of \(x\), which are the solutions to the equation.
The formula contains a "+" and a "-" sign, representing the two potential solutions for \(x\). This feature highlights the power of the quadratic formula in solving quadratic equations with ease. The two solutions are sometimes called the roots of the equation.

Discriminant

The discriminant is a key part of the quadratic formula found under the square root: \(b^2 - 4ac\). It helps determine the nature and number of solutions for the quadratic equation with the following insights:

• If the discriminant is greater than zero, the quadratic equation has two distinct real solutions.
• If it is equal to zero, there is exactly one real solution, also called a repeated or double root.
• If the discriminant is less than zero, the solutions are complex and not real numbers.

In our exercise, the discriminant equals 121, which is positive. Therefore, the equation has two distinct real solutions. Understanding the discriminant is a great way to quickly assess the type of solutions without having to solve the entire equation.

Standard Quadratic Form

The standard quadratic form of an equation is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a\) should not be zero.
This form ensures the equation is set up correctly for applying the quadratic formula. In the exercise, we ensured the equation was in the standard form: \(3x^2 - 13x + 4 = 0\).
The constants here are:\(a = 3\), \(b = -13\), and \(c = 4\).
Starting with the equation in this form is essential for correctly identifying the coefficients needed to apply the quadratic formula.

Solutions of a Quadratic Equation

Solutions to a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).
After using the quadratic formula and solving the equation step-by-step, our exercise shows that the solutions to the specific equation \(3x^2 - 13x + 4 = 0\) are:

• \(x = 4\)
• \(x = \frac{1}{3}\)

These solutions indicate the points where the corresponding quadratic function \(y = ax^2 + bx + c\) intersects the x-axis on a graph.
It means the polynomial has two x-intercepts, highlighting the important role solutions play in the interpretation of quadratic equations.