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Chapter 17: Problem 527

Determine the character of the roots of the equation $$ 4 x^{2}-12 x+9=0 $$

Short Answer

Expert verified
The character of the roots of the equation \(4x^2 - 12x + 9 = 0\) is real and equal.

Step by step solution

01

Identify the coefficients a, b, and c

For the given equation, we have a quadratic in the form: \(ax^2 + bx + c = 0\) Comparing it with the given equation: $$ 4x^2 - 12x + 9 = 0 $$ we have: a = 4, b = -12, and c = 9
02

Calculate the discriminant 螖

Now, let's calculate the discriminant using the formula: 螖 = b^2 - 4ac Plugging the values of a, b, and c: 螖 = (-12)^2 - 4(4)(9) 螖 = 144 - 144 螖 = 0
03

Determine the character of the roots based on 螖

Since the discriminant 螖 = 0, we can infer that the roots of the given quadratic equation are real and equal. So, the character of the roots of the equation \(4x^2 - 12x + 9 = 0\) is real and equal.

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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 fundamental components of algebra, often represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). These equations are characterized by their highest degree, which is 2, meaning the largest exponent of the variable \(x\) is two.

When faced with a quadratic equation, the primary goal is typically to solve for the values of \(x\) that make the equation true, also known as the 'roots' or 'solutions' of the equation. These roots can be found through various methods, such as factoring, completing the square, the quadratic formula, or graphing.
Discriminant
The discriminant is a very powerful concept when dealing with quadratic equations. It is represented by the Greek letter Delta (螖) and derived from the coefficients of the quadratic equation: \( 螖 = b^2 - 4ac \).

The value of the discriminant gives us vital information about the nature of the roots of the quadratic equation without actually solving it:
  • If \(螖 > 0\), there are two distinct real roots.
  • If \(螖 = 0\), there is one repeated real root, also known as a double root.
  • If \(螖 < 0\), there are no real roots; instead, there are two complex roots which are complex conjugates of each other.

This concept not only allows us to understand the roots' nature but can also be a quick way to determine the possibility of factoring a quadratic with real numbers.
Real and Equal Roots
In the context of quadratic equations, when we say the roots are real and equal, it means that the equation has a single solution for \(x\) which is repeated. This happens when the discriminant (螖) equals zero. The single root can be found using the formula \(x = -b/(2a)\).

In our example, \(4x^2 - 12x + 9 = 0\), we calculated that the discriminant is zero. This tells us the roots are real (not complex) and equal (the same value of \(x\) works twice). Therefore, this quadratic equation intercepts the \(x\)-axis at one point, and the vertex of the parabola that it represents coincides with this interception point on the graph.

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Most popular questions from this chapter

Show that the graph of \(\mathrm{y}=-\mathrm{x}^{2}+\mathrm{x}-1\) has no real zeros.

Without solving the equation \(2 x^{2}-3 x+5=0\) determine the nature of its roots.

Compute the value of the discriminant and then determine the nature of the roots of each of the following four equations: $$ \begin{array}{r} 4 x^{2}-12 x+9=0 \\ 3 x^{2}-7 x-6=0 \\ 5 x^{2}+2 x-9=0 \\ \text { and } \quad x^{2}+3 x+5=0 \end{array} $$

Find the equation whose roots are \(3+\sqrt{2}\) and \(3-\sqrt{2}\).

Solve for the roots of the equation \(6 \mathrm{x}^{2}+5 \mathrm{x}-2=0\) and for the roots of the equation \(3 x^{2}+4 x-4=0\)
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