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Chapter 4: Problem 55

Find the roots of the given functions. \(f(x)=x^{2}+10 x-24\)

Short Answer

Expert verified
The roots are \(x = 2\) and \(x = -12\).

Step by step solution

01

Identify Function Components

The given function is a quadratic equation of the form \(ax^2 + bx + c\), where \(a = 1\), \(b = 10\), and \(c = -24\).
02

Determine the Discriminant

Calculate the discriminant \(D\) using the formula \(D = b^2 - 4ac\).Substitute the values of \(a\), \(b\), and \(c\): \[D = 10^2 - 4(1)(-24) = 100 + 96 = 196\]
03

Calculate the Roots Using the Quadratic Formula

Since the discriminant is positive, there are two real roots. Use the quadratic formula: \[x = \frac{-b \pm \sqrt{D}}{2a}\] Substitute \(a = 1\), \(b = 10\), and \(D = 196\) into the formula: \[x = \frac{-10 \pm \sqrt{196}}{2}\]Simplify by calculating the square root: \[x = \frac{-10 \pm 14}{2}\]
04

Simplify the Results for Both Roots

Calculate both possible roots by solving the two results from the formula:1. \(x = \frac{-10 + 14}{2} = \frac{4}{2} = 2\)2. \(x = \frac{-10 - 14}{2} = \frac{-24}{2} = -12\)The roots of the equation are \(x = 2\) and \(x = -12\).

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Key Concepts

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

Discriminant
The discriminant is a key concept when working with quadratic equations. It gives us important information about the nature of the roots without actually solving the equation.
To find the discriminant, use the formula:
  • The equation must be in the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients.
  • The discriminant \( D \) is calculated using the expression \( D = b^2 - 4ac \).
The value of \( D \) determines the nature of the roots:
  • If \( D > 0 \), the equation has two distinct real roots.
  • If \( D = 0 \), there is exactly one real root, also known as a repeated or double root.
  • If \( D < 0 \), the roots are not real; they are complex or imaginary.
In our exercise, \( D = 196 \) was positive, indicating two real roots for the quadratic equation \( x^2 + 10x - 24 \).
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of any quadratic equation, even if they aren't easily factorable. The formula is given by: \[ x = \frac{-b \pm \sqrt{D}}{2a} \]Let's break this down:
  • \( b \) is the coefficient of the \( x \) term,
  • \( D \) is the discriminant (\( b^2 - 4ac \)),
  • \( 2a \) is twice the coefficient of \( x^2 \).
To find the roots using this formula, simply plug the values you have into the formula.
For our equation, we substituted \( a = 1 \), \( b = 10 \), and \( D = 196 \):
  • \( x = \frac{-10 \pm \sqrt{196}}{2} \),
The phrase "\( \pm \)" means you'll calculate two separate solutions: one with addition and one with subtraction. The final step involves solving two equations to get the two roots of the quadratic.
This yields \( x = 2 \) and \( x = -12 \).
Real Roots
Real roots refer to the solutions of a quadratic equation that are actual numbers as opposed to complex numbers. Whether a quadratic equation produces real roots depends primarily on the value of its discriminant.
To identify the nature of the roots:
  • When the discriminant \( D > 0 \), the roots are distinct and real. There are two different real numbers that satisfy the equation.
  • When \( D = 0 \), there is one real root, which means the solution will be a repeated value.
  • When \( D < 0 \), the roots are complex, meaning they don't appear on the number line as real numbers.
In our example, since \( D = 196 \), this confirmed that the quadratic equation \( x^2 + 10x - 24 \) has two distinct real roots: \( x = 2 \) and \( x = -12 \). Understanding real roots is essential for solving and interpreting the solutions of quadratic equations in real-world contexts.

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