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Chapter 6: Problem 38

Find the zeros of each function. $$ f(x)=6 x^{2}+17 x+6 $$

Short Answer

Expert verified
The zeros are \( x = \frac{-17 + \sqrt{145}}{12} \) and \( x = \frac{-17 - \sqrt{145}}{12} \).

Step by step solution

01

- Identify the Coefficients

For the quadratic function \( f(x) = 6x^2 + 17x + 6 \), the coefficients are: \( a = 6 \), \( b = 17 \), and \( c = 6 \).
02

- Start with the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the coefficients into the formula: \( a = 6 \), \( b = 17 \), \( c = 6 \).
03

- Calculate the Discriminant

The discriminant is given by \( b^2 - 4ac \). Substituting the values, \( 17^2 - 4 \cdot 6 \cdot 6 = 289 - 144 = 145 \).
04

- Substitute into the Quadratic Formula

Substitute \( b = 17 \), \( a = 6 \), and the discriminant \( \sqrt{145} \) into the quadratic formula: \[ x = \frac{-17 \pm \sqrt{145}}{12} \].
05

- Write the Final Answer

The zeros of the function are: \( x = \frac{-17 + \sqrt{145}}{12} \) and \( x = \frac{-17 - \sqrt{145}}{12} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic formula
The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is typically in the form of \(ax^2 + bx + c = 0\). The formula itself is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula helps us find the values of \(x\) that make the equation equal to zero. These values are the 'roots' or 'zeros'. The \(b^2 - 4ac\) part of the formula is called the discriminant, and it tells us key information about the number and type of roots we can expect.
coefficients
In the quadratic equation \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are known as coefficients. They are constant numbers that tell us about the shape and position of the parabola when plotted on a graph. For example, in the equation \(6x^2 + 17x + 6\):
  • \(a = 6\), which is the coefficient of \(x^2\)
  • \(b = 17\), which is the coefficient of \(x\)
  • \(c = 6\), which is the constant term.
Identifying these coefficients is the first step when using the quadratic formula.
discriminant
The discriminant is a part of the quadratic formula given by \(b^2 - 4ac\). It provides important information about the nature of the zeros of the quadratic function. You substitute the values of \(a\), \(b\), and \(c\) into \(b^2 - 4ac\) to find the discriminant.
For the function \(6x^2 + 17x + 6\), the discriminant is calculated as \(17^2 - 4 \cdot 6 \cdot 6 = 289 - 144 = 145\).
The value of the discriminant reveals:
  • If it is positive, there are two distinct real roots.
  • If it is zero, there is one real root.
  • If it is negative, there are no real roots (the roots are complex numbers).
In our example, the discriminant is 145, a positive number, which means the function has two distinct real roots.
zeros of a function
The zeros of a function are the points where the graph of the function intersects the x-axis. These are the values of \(x\) that make the function equal to zero. For the quadratic function \(6x^2 + 17x + 6\), we have already identified the coefficients \(a = 6\), \(b = 17\), and \(c = 6\) and calculated the discriminant as 145.
Using the quadratic formula, the zeros are calculated as: \[ x = \frac{-17 \pm \sqrt{145}}{12}\]
This means the zeros are \(x = \frac{-17 + \sqrt{145}}{12}\) and \(x = \frac{-17 - \sqrt{145}}{12}\).
These zeros tell us the x-values where our quadratic function equals to zero, and hence, where it crosses the x-axis.

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