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Chapter 3: Problem 85

Find the zeros of the function algebraically. Give exact answers. $$f(x)=4 x^{2}+3 x-3$$

Short Answer

Expert verified
The zeros of the function are \( \frac{-3 + \sqrt{57}}{8} \) and \( \frac{-3 - \sqrt{57}}{8} \).

Step by step solution

01

Identify the Quadratic Function

The given function is a quadratic function in the form of \[ f(x) = ax^2 + bx + c \]Here, identify the coefficients: \( a = 4 \), \( b = 3 \), and \( c = -3 \).
02

Apply the Quadratic Formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the identified coefficients \( a, b, \) and \( c \) into the formula: \( a = 4 \), \( b = 3 \), and \( c = -3 \).
03

Calculate the Discriminant

The discriminant (\( \Delta \)) is defined as \( \Delta = b^2 - 4ac \). Substitute \( a, b, \) and \( c \) into the discriminant: \[ \Delta = 3^2 - 4(4)(-3) \] \[ \Delta = 9 + 48 \] \[ \Delta = 57 \]
04

Substitute Back Into the Quadratic Formula

Using the discriminant calculated in Step 3, substitute back into the quadratic formula: \[ x = \frac{-3 \pm \sqrt{57}}{8} \].
05

Simplify the Solutions

Thus, the exact zeros of the function are: \[ x_1 = \frac{-3 + \sqrt{57}}{8} \] and \[ x_2 = \frac{-3 - \sqrt{57}}{8} \].

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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 used to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). Quadratic equations include terms like \(x^2\), \(x\), and a constant. The quadratic formula provides a straightforward method to find the solutions or 'roots' of these equations, which are the values of \(x\) where the equation equals zero.
The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula may look a bit complex, but each part has a specific role:
  • The term \(-b\) gives the opposite of the coefficient of \(x\).
  • The term \(\b^2 - 4ac\) under the square root is called the discriminant.
  • The \(2a\) in the denominator adjusts for the coefficient of \(x^2\).
To use this formula effectively, you need to correctly identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation and plug them into the formula.
Discriminant
The discriminant is a key part of the quadratic formula, and it tells us about the nature of the roots of the quadratic equation. The discriminant is given by the expression: \[ \Delta = b^2 - 4ac \] The value of the discriminant (\Delta) can help determine the nature of the roots:
  • If \Delta > 0, there are two distinct real roots.
  • If \Delta = 0, there is exactly one real root, often called a repeated or double root.
  • If \Delta < 0, there are no real roots; instead, there are two complex roots.
In the given problem, we calculated the discriminant as: \[ \Delta = 3^2 - 4(4)(-3) = 9 + 48 = 57 \] This positive value tells us that the quadratic equation has two distinct real roots.
Solving Quadratic Equations
Solving quadratic equations involves finding the values of \(x\) that make the equation equal to zero. Here's a clear step-by-step method using the quadratic formula:
  1. Identify the coefficients: For the equation \(4x^2 + 3x - 3 = 0\), the coefficients are \(a = 4\), \(b = 3\), and \(c = -3\).
  2. Calculate the discriminant: Plug these values into the discriminant formula \( \Delta = b^2 - 4ac \) to get \(\Delta = 57\).
  3. Substitute into the quadratic formula: Use the values of \(a\), \(b\), and \( \Delta \) in the quadratic formula \[\backslash x = \frac{-3 \pm \sqrt{57}}{8} \] to find the solutions.
  4. Simplify the solutions: The solutions are: \[ x_1 = \frac{-3 + \sqrt{57}}{8} \] and \[ x_2 = \frac{-3 - \sqrt{57}}{8} \].
Following these steps systematically ensures you will find the exact roots of any quadratic equation. Practice using the quadratic formula with different sets of coefficients to become more comfortable with this method.

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