Question

Problems 112-121 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the real zeros of \(f(x)=3 x^{2}-10 x+5\)

Short answer

The real zeros are \( x = \frac{5 + \sqrt{10}}{3} \) and \( x = \frac{5 - \sqrt{10}}{3} \).

Step-by-step solution

- Write down the quadratic equation

Start by writing the given quadratic function in standard form: \[ f(x) = 3x^2 - 10x + 5 \]

- Determine coefficients for the quadratic formula

Identify the coefficients in the quadratic equation: \( a = 3 \), \( b = -10 \), and \( c = 5 \)

- Write down the quadratic formula

Recall the quadratic formula to find the zeros of a quadratic equation ax^2 + bx + c = 0: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

- Substitute coefficients into the quadratic formula

Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula: \[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 5}}{2 \cdot 3} \] Simplify inside the square root: \[ x = \frac{10 \pm \sqrt{100 - 60}}{6} \] \[ x = \frac{10 \pm \sqrt{40}}{6} \]

- Simplify the expression under the square root

Simplify \( \sqrt{40} \) as: \( \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} \), so: \[ x = \frac{10 \pm 2\sqrt{10}}{6} \]

- Simplify the final expression

Simplify the fraction: \[ x = \frac{10}{6} \pm \frac{2\sqrt{10}}{6} \] \[ x = \frac{5}{3} \pm \frac{\sqrt{10}}{3} \]

- Write the final solutions

The solutions to the quadratic equation are: \[ x = \frac{5 + \sqrt{10}}{3} \] and \[ x = \frac{5 - \sqrt{10}}{3} \]

Key concepts

Real Zeros

Finding the real zeros of a function means identifying the x-values where the function equals zero. It's like asking, 'Where does the graph of the function touch or cross the x-axis?' For a quadratic function like \( f(x) = 3x^2 - 10x + 5 \), its real zeros are the solutions for \( x \) when \( f(x) = 0 \). These zeros can be found using various methods, such as factoring, completing the square, or using the quadratic formula. When the zeros are real numbers, they can be plotted on the number line or graph to show where the function has no value. In our exercise, finding real zeros involves using the quadratic formula, as the equation may not factorize easily.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Here’s how it works:

• Start by identifying the coefficients \( a \), \( b \), and \( c \) from the equation.
• Substitute these coefficients into the formula.
• Calculate the discriminant, \( b^2 - 4ac \), to determine the nature of the roots.

The discriminant helps to decide whether the roots are real or complex. If \( b^2 - 4ac \) is positive, the quadratic equation has two distinct real roots. If it's zero, there is one real root (repeated). In our step-by-step solution, the discriminant \( 100 - 60 = 40 \) is a positive number, indicating two real solutions.

Simplifying Expressions

Simplifying expressions involves making the equation as simple as possible. This could mean reducing fractions, combining like terms, or simplifying roots. Within the quadratic formula, you may need to simplify the term inside the square root (the discriminant) and the resulting expressions for \( x \). For instance, after substituting into the quadratic formula: \[ x = \frac{10 \pm \sqrt{40}}{6} \], we simplify \( \sqrt{40} \) as follows: \( \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} \). So, we get: \[ x = \frac{10 \pm 2\sqrt{10}}{6} \]. Further simplification leads to: \[ x = \frac{5}{3} \pm \frac{\sqrt{10}}{3} \]. Breaking down and simplifying each step ensures the final solution is in its simplest form, making it easier to understand and work with.

Coefficients

In any quadratic equation \( ax^2 + bx + c = 0 \), the coefficients \( a \), \( b \), and \( c \) are crucial. They define the shape and position of the quadratic graph (parabola) and are essential in solving the equation.

• The coefficient \( a \) affects the width and direction of the parabola. If \( a \) is positive, the parabola opens upwards. If negative, it opens downwards.
• The coefficient \( b \) determines the position of the vertex along the x-axis.
• The coefficient \( c \) represents the y-intercept, where the parabola crosses the y-axis.

For our quadratic equation \( 3x^2 - 10x + 5 \), we have \( a = 3 \), \( b = -10 \), and \( c = 5 \). Understanding these coefficients and their roles helps to predict the behavior of the quadratic function and is fundamental in applying the quadratic formula accurately.