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Chapter 6: Problem 54

Find the zeros of each function. $$ g(x)=3 x^{2}-21 x+30 $$

Short Answer

Expert verified
The zeros are 5 and 2.

Step by step solution

01

Identify the quadratic equation standard form

The given function is in the form of a quadratic equation: \[ g(x) = 3x^2 - 21x + 30 \]. Here, identify the coefficients where: \( a = 3 \), \( b = -21 \), and \( c = 30 \).
02

Apply the quadratic formula

To find the zeros of the function, use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substitute the coefficients \( a = 3 \), \( b = -21 \), and \( c = 30 \) into the formula.
03

Calculate the discriminant

Calculate the discriminant \( \Delta \) using the formula: \[ \Delta = b^2 - 4ac \]. Substitute the values \( b = -21 \), \( a = 3 \), and \( c = 30 \): \[ \Delta = (-21)^2 - 4(3)(30) = 441 - 360 = 81 \].
04

Solve for the roots

Now that the discriminant \( \Delta = 81 \), which is a perfect square, find the roots: \[ x = \frac{21 \pm \sqrt{81}}{6} \]. This simplifies to: \[ x = \frac{21 \pm 9}{6} \].
05

Find the two solutions

Calculate the two possible values for \( x \): \[ x = \frac{21 + 9}{6} = \frac{30}{6} = 5 \] and \[ x = \frac{21 - 9}{6} = \frac{12}{6} = 2 \]. Thus, the zeros of the function are 5 and 2.

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Key Concepts

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

Quadratic Formula
To find the zeros of a quadratic function, one of the most reliable methods is the quadratic formula. Quadratic equations are in the form \[ax^2 + bx + c = 0\]. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula helps you find the values of x that make the equation equal to zero. Always start by identifying the coefficients: a, b, and c. Plug these into the formula and simplify step-by-step. For example, in the function \[g(x) = 3x^2 - 21x + 30\], we have \[a = 3\], \[b = -21\], and \[c = 30\]. By substituting these values into the quadratic formula, you will be on your way to solving for the zeros of the function.
Discriminant
The discriminant is a crucial part of the quadratic formula. It is located under the square root in the formula and is computed as \[ \Delta = b^2 - 4ac \]. The value of the discriminant determines the nature of the roots of a quadratic equation:
  • If \( \Delta > 0 \), there are two distinct real roots.
  • If \( \Delta = 0 \), there is one real root.
  • If \( \Delta < 0 \), there are no real roots, but two complex roots.
For example, in \[g(x) = 3x^2 - 21x + 30\], we calculate the discriminant as \[ \Delta = (-21)^2 - 4(3)(30) = 441 - 360 = 81 \]. Since 81 is greater than zero, the function has two distinct real roots.
Zeros of a Function
The zeros of a function are the x-values at which the function equals zero. They are also known as the roots or solutions of the equation. For a quadratic function, finding the zeros means solving the equation \[ax^2 + bx + c = 0\].

Using the quadratic formula, we already calculated our discriminant as 81. Now, find the roots by substituting this into the formula: \[ x = \frac{21 \pm \sqrt{81}}{6} \]. This simplifies to \[ x = \frac{21 + 9}{6} \] and \[ x = \frac{21 - 9}{6} \]. Calculating these, we get two roots: \[ x = 5 \] and \[ x = 2 \]. Thus, the zeros of \[g(x) = 3x^2 - 21x + 30\] are 5 and 2. These are the values where the function intersects the x-axis.

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