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Chapter 1: Problem 73

Quadratic Equations Find all real solutions of the quadratic equation. $$9 x^{2}+12 x+4=0$$

Short Answer

Expert verified
The equation has one real solution: \( x = -\frac{2}{3} \).

Step by step solution

01

Identify coefficients

The quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, \( a = 9 \), \( b = 12 \), and \( c = 4 \).
02

Calculate the discriminant

The discriminant \( \Delta \) is calculated as \( b^2 - 4ac \). Substituting the coefficients, we have \( \Delta = 12^2 - 4 \times 9 \times 4 = 144 - 144 = 0 \).
03

Determine the number and type of solutions

Since the discriminant \( \Delta = 0 \), this indicates that there is exactly one real solution for the quadratic equation.
04

Solve for the root

When \( \Delta = 0 \), the quadratic has a double (repeated) root given by \( x = -\frac{b}{2a} \). Plug in the values: \( x = -\frac{12}{2 \times 9} = -\frac{12}{18} = -\frac{2}{3} \).

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Key Concepts

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

Discriminant
The discriminant is a key component when solving quadratic equations. It's found inside the quadratic formula and helps determine the nature of the roots of the equation. It's calculated using the formula:
  • \( \Delta = b^2 - 4ac \)
Here, \( a \), \( b \), and \( c \) are the coefficients of the terms in the quadratic equation \( ax^2 + bx + c = 0 \). The value of the discriminant reveals important information:
  • If \( \Delta > 0 \), there are two distinct real solutions.
  • If \( \Delta = 0 \), there is exactly one real solution, known as a double root.
  • If \( \Delta < 0 \), there are no real solutions, but two complex solutions.
In our exercise, the discriminant \( \Delta = 0 \), indicating one real solution.
Real Solutions
Real solutions of a quadratic equation refer to the values of \( x \) that satisfy the equation, resulting in zero. When solving quadratics, it's important to determine whether the solutions are real or complex. Real solutions mean the graph of the equation intersects the x-axis.

For a quadratic equation like \( 9x^2 + 12x + 4 = 0 \), real solutions can manifest in different ways:
  • Two distinct real solutions occur when the discriminant is greater than zero.
  • Only one real solution, a double root, appears when the discriminant is zero.
Graphically, distinct real solutions intersect the x-axis at two points, while a double root "touches" the x-axis at one point. In our problem, since \( \Delta = 0 \), the solution is a single point, \( x = -\frac{2}{3} \).
Double Root
A double root happens when a quadratic equation has exactly one real solution due to the discriminant being zero. It occurs when the quadratic's graph is tangent to the x-axis, touching it at exactly one point but not crossing it.

In the equation \( 9x^2 + 12x + 4 = 0 \), the repeated solution is calculated using the formula:
  • \( x = -\frac{b}{2a} \)
Substitute the values from the equation: \( a = 9 \), \( b = 12 \).
Calculate the double root:
  • \( x = -\frac{12}{2 \times 9} = -\frac{12}{18} = -\frac{2}{3} \)
The quadratic equation's graph merely touches the x-axis at \( x = -\frac{2}{3} \), showcasing the single but repeated nature of this root. This ensures the quadratic doesn't "open up" or "down" through the x-axis at this point.

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Most popular questions from this chapter

A determined gardener has \(120 \mathrm{ft}\) of deer-resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area that is enclosed to be at least \(800 \mathrm{ft}^{2}\). What range of values is possible for the length of her garden?

Simplify the expression. (a) \(\left(8 y^{3}\right)^{-2 / 3}\) (b) \(\left(u^{4} v^{6}\right)^{-1 / 3}\)

Solve the equation for the variable \(x\). The constants \(a\) and \(b\) represent positive real numbers. $$a^{3} x^{3}+b^{3}=0$$

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\left(7.2 \times 10^{-9}\right)\left(1.806 \times 10^{-12}\right)$$

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt{s \sqrt{s^{3}}}\) (b) \(\sqrt[3]{\frac{54 x^{2} y^{4}}{2 x^{5} y}}\)
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