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Chapter 0: Problem 104

Solve the quadratic equation by the method of your choice. $$9-6 x+x^{2}=0$$

Short Answer

Expert verified
The quadratic equation \(9-6 x+x^{2}=0\) has only one distinct solution, which is \(x = 3\).

Step by step solution

01

Rearrange to Standard Form

The standard form of a quadratic equation is \(a x^2 + bx + c = 0\). So, rearrange the equation \(9-6 x+x^{2}=0\) to standard form, which is \(x^{2}-6x+9=0\) with \(a = 1\), \(b = -6\), and \(c = 9\)
02

Apply the Quadratic Formula

The quadratic formula is \((-b卤\sqrt{b^2-4ac})/2a\). Apply this formula by substituting \(a = 1\), \(b = -6\), and \(c = 9\). This yields two roots, \(x1\) and \(x2\), calculated as follows: \(x1 = (-(-6) + \sqrt{(-6)^2 - 4*1*9}) / (2*1)\) and \(x2 = (-(-6) - \sqrt{(-6)^2 - 4*1*9}) / (2*1)\). Simplified, this results in \(x1 = x2 = 3\)
03

Interpret the Solution

Since the solutions are the same, it means that the quadratic equation has only one distinct root, which in this case, is \(x = 3\). This also signifies that the graph of the quadratic equation will touch the x-axis at one point only, at \(x = 3\)

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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 find the solutions (or "roots") of any quadratic equation鈥攕tandard form or not. It provides a straightforward method for calculating the roots regardless of whether they are real or complex numbers. Here is what the formula looks like: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
  • The symbol \( \pm \) indicates there could be two solutions for \( x \), one involving addition and the other involving subtraction.

  • \( a \), \( b \), and \( c \) are coefficients from the standard form of the quadratic equation \( ax^2 + bx + c = 0 \).
  • This formula originates from the process of completing the square, a method used to solve quadratics by transforming them into a perfect square trinomial.
To solve a quadratic using this formula, simply substitute the coefficients \( a \), \( b \), and \( c \) from your equation into the formula and compute. It's efficient and covers all cases: real, repeated, or complex roots.
Standard Form of a Quadratic Equation
Before solving a quadratic equation, it must be in standard form, which is written as \( ax^2 + bx + c = 0 \). This structure is important as it prepares the equation for various solution methods, such as factoring, completing the square, or using the quadratic formula. Here鈥檚 a breakdown:
  • \( a \) is the coefficient of the \( x^2 \) term. It must not be zero to ensure the equation is quadratic.

  • \( b \) is the coefficient of the \( x \) term. It can be zero, in which case the equation becomes a simple square.
  • \( c \) is the constant term. It can shift the graph of the equation up or down.

For the given equation \(9 - 6x + x^2 = 0\), rearranging it yields the standard form \(x^2 - 6x + 9 = 0\). Recognizing and arranging equations in this form is a critical first step in solving them using the quadratic formula or other methods.
Roots of a Quadratic Equation
The roots of a quadratic equation are the values of \( x \) that make the equation true, effectively providing points where the graph of the equation intersects the x-axis. Understanding roots is crucial since:
  • When you use the quadratic formula, you find these roots as solutions to the equation.

  • There can be two distinct roots, one repeated root (resulting from perfect squares), or complex roots (when discriminant \( b^2 - 4ac \) is negative).
In the exercise provided, solving the quadratic \(x^2 - 6x + 9 = 0\) results in a single, repeated root of \( x = 3 \). This means the graph of the equation is tangent to the x-axis at this point, a scenario that happens when the discriminant is zero, indicating a perfect square trinomial. Finding and interpreting these roots lets us understand the real-life scenarios represented by quadratic models, from projectile motion to optimization problems.

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Most popular questions from this chapter

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and \(2012 .\) Also shown is the percentage of households in which a person of faith is married to someone with no religion. GRAPH CAN'T COPY. The formula $$I=\frac{1}{4} x+26$$ models the percentage of U.S. households with an interfaith marriage, \(I, x\) years after \(1988 .\) The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, \(N, x\) years after \(1988 .\) Use these models to solve Exercises \(107-108\). a. In which years will more than \(34 \%\) of U.S. households have an interfaith marriage? b. In which years will more than \(15 \%\) of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than \(34 \%\) of households have an interfaith marriage and more than \(15 \%\) have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than \(34 \%\) of households have an interfaith marriage or more than \(15 \%\) have a faith/no religion marriage?

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{4} \cdot \frac{8}{15}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The cube root of \(-8\) is not a real number.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathbf{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

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