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Chapter 11: Problem 5

Use the quadratic formula to solve each equation. These equations have real number solutions only. See Examples I through 3. $$ x^{2}-6 x+9=0 $$

Short Answer

Expert verified
The solution to the equation is \( x = 3 \).

Step by step solution

01

Identify the Quadratic Equation

The given equation is \( x^2 - 6x + 9 = 0 \). This equation is in the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -6 \), and \( c = 9 \).
02

Recall the Quadratic Formula

The quadratic formula for solving equations of the form \( ax^2 + bx + c = 0 \) is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
03

Substitute Values into the Quadratic Formula

Substitute \( a = 1 \), \( b = -6 \), and \( c = 9 \) into the quadratic formula: \( x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} \).
04

Simplify the Expression

First, compute inside the square root: \( (-6)^2 = 36 \) and \( 4 \cdot 1 \cdot 9 = 36 \). Therefore, \( b^2 - 4ac = 36 - 36 = 0 \). Substitute into the formula: \( x = \frac{6 \pm \sqrt{0}}{2} \).
05

Calculate the Final Result

Since \( \sqrt{0} = 0 \), the equation simplifies to \( x = \frac{6}{2} \). This gives \( x = 3 \). Since the discriminant \( b^2 - 4ac \) is zero, there is only one unique solution.

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Key Concepts

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

Quadratic Equations
Quadratic equations are a fundamental part of algebra and are characterized by the presence of an unknown variable squared. These equations typically appear in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the unknown value you need to solve for.
To tackle these equations, it's crucial to identify all the components properly:
  • \( a \): The coefficient of \( x^2 \). It determines the "width" and "direction" of the parabola when graphing the equation.
  • \( b \): The coefficient of \( x \). It affects the position of the parabola along the x-axis.
  • \( c \): The constant term. It shows where the parabola intersects the y-axis.
Understanding these elements helps us not just to solve quadratics, but also visualize how they behave in a graph, facilitating better prediction and interpretation of the solutions.
Solving Equations
When it comes to solving quadratic equations, the quadratic formula is a powerful tool. The quadratic formula is\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]and it provides the solutions to any quadratic equation by substituting the values of \( a \), \( b \), and \( c \) from the equation into this formula.
To use the formula:
  • Identify the values of \( a \), \( b \), and \( c \) from the equation.
  • Substitute these values into the formula.
  • Calculate inside the square root, known as the discriminant.
  • Simplify the expression to find the value(s) of \( x \).
A good example involves an equation where \( a = 1 \), \( b = -6 \), and \( c = 9 \). Substituting these into the formula allows for calculating the solution(s) efficiently, by focusing on the discriminant value.
Discriminant
The discriminant in the quadratic formula is the expression \( b^2 - 4ac \), found inside the square root. It plays a critical role in determining the nature of the roots of the quadratic equation.
Understanding the discriminant helps to quickly establish:
  • If the discriminant is positive, \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
  • If the discriminant equals zero, \( b^2 - 4ac = 0 \), there is exactly one real root, indicating a repeated or double root.
  • If the discriminant is negative, \( b^2 - 4ac < 0 \), the equation has no real roots, instead two complex conjugate roots.
In the given problem, the discriminant is zero, leading to exactly one real solution. This matches our final result of \( x = 3 \), confirming that our solution approach was both efficient and accurate.

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Most popular questions from this chapter

Sketch the graph of each function. See Section 11.5 $$ f(x)=(x+1)^{2}+4 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ g(x)=(x-6)^{2}+1 $$

Sketch the graph of each function. See Section 11.5 $$ f(x)=x^{2}+2 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ g(x)=(x+2)^{2}-5 $$

Sketch the graph of each function. See Section 11.5 $$ h(x)=x-3 $$
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