/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 65 \(f(x)=x^{2}-7 x-1\)... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 65

\(f(x)=x^{2}-7 x-1\)

Short Answer

Expert verified
The roots of the quadratic equation \(f(x) = x^{2} - 7x - 1\) are \(x_{1}=\frac{7+\sqrt{53}}{2}\) and \(x_{2}=\frac{7-\sqrt{53}}{2}\).

Step by step solution

01

Identify the coefficients

From the given equation \(f(x) = x^{2} - 7x - 1\), we can identify that: \(a = 1\), \(b = -7\), and \(c = -1\).
02

Calculate the discriminant

The discriminant \(D\) in the quadratic formula is calculated as \(D = b^{2}-4ac\). Substituting the identified values we have: \(D=(-7)^{2}-4(1)(-1) = 49 + 4 = 53\).
03

Substitute the coefficients in the quadratic formula

Now substitute the values of \(a\), \(b\), and \(D\) in the quadratic formula: \(x=\frac{-(-7)\pm\sqrt{53}}{2(1)} = \frac{7\pm\sqrt{53}}{2}\). Thus, the roots of the given quadratic equation are \(x_{1}=\frac{7+\sqrt{53}}{2}\) and \(x_{2}=\frac{7-\sqrt{53}}{2}\).

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Key Concepts

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

Understanding the Discriminant
The discriminant is a crucial part of solving quadratic equations as it determines the nature of the roots. Found in the quadratic equation formula, it is represented by \( D = b^2 - 4ac \). Here, \( b \), \( a \), and \( c \) are coefficients from the equation \( ax^2 + bx + c = 0 \). The value of the discriminant gives insights into the roots:
  • When \( D > 0 \), the equation has two distinct real roots.
  • When \( D = 0 \), there is exactly one real root, also known as a repeated or double root.
  • When \( D < 0 \), the equation has two complex roots.
Knowing this helps predict the solution's nature even before finding the specific roots, adding a strategic layer to solving quadratic equations.
Using the Quadratic Formula
The quadratic formula is a powerful tool for finding the roots of any quadratic equation. Once the coefficients \( a \), \( b \), and \( c \) are identified, the formula \( x = \frac{-b \pm \sqrt{D}}{2a} \) can be used. Here, \( D \) represents the discriminant previously calculated.
  • The symbol \( \pm \) indicates that there will be two potential solutions, as every square root operation offers both a positive and negative result.
  • You substitute the coefficients directly into the formula, solving the equation step by step.
  • This method is universally applicable, making it a go-to approach when factoring isn't straightforward.
The two solutions provided by the quadratic formula are the roots of the equation, reflecting where the parabola intersects the x-axis on a graph.
Roots of Quadratic Equations
The roots of a quadratic equation, also known as the solutions, are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). They represent the x-coordinates where the graph of the quadratic function intersects the x-axis in a coordinate plane.
  • If the roots are real and distinct (\( D > 0 \)), they appear as two separate points on the x-axis.
  • If there's a single real root (\( D = 0 \)), the vertex of the parabola touches the x-axis at just one point. This is a tangent point, showing where the symmetrical, U-shaped curve just "kisses" the axis.
  • If the roots are complex (\( D < 0 \)), they do not intersect with the x-axis at all, and the curve lies entirely above or below it.
Understanding the nature of these roots is essential in solving and graphing quadratic equations, providing a complete picture of a quadratic function's behavior.

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