Question

Find the roots of each quadratic by any of the methods shown in this section. Keep three significant digits. For some, use more than one method and compare results. Explicit Functions. $$x^{2}-6 x+7=0$$

Short answer

The roots of the quadratic equation \( x^2 - 6x + 7 = 0 \) are approximately \( x \approx 4.41 \) and \( x \approx 1.59 \) to three significant digits.

Step-by-step solution

Determine if Factorization is Possible

Check if the quadratic equation can be easily factored into two binomials. In this case, the equation is in the form of \( x^2 - 6x + 7 = 0 \). Since the factors of 7 that sum up to -6 are not immediately apparent, it may be hard or impossible to factor this expression neatly. Thus, we should try another method such as the quadratic formula or completing the square.

Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots of the equation, where \( a = 1 \), \( b = -6 \), and \( c = 7 \). Substitute these values into the formula to evaluate.

Calculate the Discriminant

Calculate the discriminant \( \Delta = b^2 - 4ac \) to determine the nature of the roots. In this case, \( \Delta = (-6)^2 - 4(1)(7) = 36 - 28 = 8 \). Since the discriminant is positive, we know there are 2 real roots, and we can proceed with the quadratic formula.

Find the Roots Using the Quadratic Formula

Substitute the values into the quadratic formula to find the roots: \( x = \frac{-(-6) \pm \sqrt{8}}{2(1)} \). Simplify the values to get \( x = \frac{6 \pm \sqrt{8}}{2} \).

Simplify the Result

Simplify the roots to their simplest form: \( x = \frac{6 \pm 2\sqrt{2}}{2} \). This simplifies further to \( x = 3 \pm \sqrt{2} \) when you divide both terms in the numerator by 2.

Write the Roots with Three Significant Digits

By evaluating \( 3 \pm \sqrt{2} \), we find the roots to be approximately \( x = 3 + \sqrt{2} \approx 4.41 \) and \( x = 3 - \sqrt{2} \approx 1.59 \) to three significant digits.

Key concepts

Quadratic Formula

When solving a quadratic equation, if factorization is challenging, the quadratic formula can be a saving grace. This powerful tool is written as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) and can solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients from the equation, with \( a \) not being zero.

To apply the quadratic formula, follow a few straightforward steps:

• Identify the coefficients \( a \) (before \( x^2 \)), \( b \) (before \( x \)), and \( c \) (the constant term).
• Substitute these values into the formula, paying attention to the signs.
• Calculate the value under the square root, known as the discriminant, which tells you about the nature of the roots - whether they're real or complex.
• Perform the operations in the formula to find the value of \( x \) in two possible versions, \( x_1 \) and \( x_2 \) by using both the plus and minus signs in front of the square root.
• Simplify the results to find the roots of the equation.

In the exercise provided, using the quadratic formula quickly provides the roots as \( x = 3 \pm \sqrt{2} \) to their simplest form.

Factorization Method

The factorization method is a technique used to solve quadratic equations that can be decomposed into two binomial expressions. The basic principle is to rewrite the quadratic in the form \( (x - p)(x - q) = 0 \) where \( p \) and \( q \) are the roots of the quadratic equation.

This method is most effective when:

• The \( a \) coefficient (before \( x^2 \) term) is 1 or can be made 1 by dividing the entire equation.
• The constant term (\( c \)) and the coefficient of \( x \) (\( b \) term) can be factored into integers that are products summing to \( b \) and multiplying to \( ac \) (if \( a \) is not 1).

Unfortunately, not all quadratic equations can be easily factored. If no apparent factors of \( ac \) sum up to \( b \) as in our exercise \( x^2 - 6x + 7 = 0 \), the factorization method may not be directly applicable without complex factorization, and alternative methods should be considered.

Discriminant

A key concept in understanding quadratic equations is the discriminant, denoted by \( \Delta \) and calculated as \( b^2 - 4ac \). The discriminant is crucial as it determines the nature of the roots of a quadratic equation:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (also called a repeated or double root).
• If \( \Delta < 0 \), there are no real roots, but rather two complex roots.

In the given exercise, the discriminant is calculated to be 8. A positive discriminant indicates that we have two distinct real roots. Since the discriminant is not a perfect square, the roots are irrational and can be simplified further but not into exact integers. Understanding the discriminant provides insight not just into the type of the roots but also lends a predictive step before even performing the whole quadratic formula operation.

Roots of Quadratic

The roots of a quadratic equation are the values of \( x \) that make the equation \( ax^2 + bx + c = 0 \) true. They are the points where the graph of the quadratic function intersects the x-axis. Roots can be real or complex and can be found using methods such as factorization, completing the square, or the quadratic formula.

In the exercise \( x^2 - 6x + 7 = 0 \), we use the quadratic formula to find the roots which are \( x = 3 \pm \sqrt{2} \) in their simplest radical form. By further simplifying, we can find that the roots to three significant digits are approximately 4.41 and 1.59. These values tell us where the parabola described by the quadratic equation crosses the x-axis, and is essential for graphing and understanding the function's behavior.