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Chapter 5: Problem 29

Solve each equation. $$ 9 x^{2}+24 x+16=0 $$

Short Answer

Expert verified
\(x = -\frac{4}{3}\)

Step by step solution

01

Identify the coefficients

Given the quadratic equation \(9x^{2}+24x+16=0\), identify the coefficients: \(a = 9\), \(b = 24\), and \(c = 16\).
02

Calculate the discriminant

Use the discriminant formula \(D = b^2 - 4ac\). Substitute the identified coefficients:\[D = 24^2 - 4(9)(16)\]Simplify the expression:\[D = 576 - 576 = 0\]The discriminant is 0, indicating a single, repeated root.
03

Find the root

Since the discriminant is 0, use the formula for repeated roots:\( x = \frac{-b}{2a} \). Substitute \(a\) and \(b\):\[ x = \frac{-24}{2(9)} = \frac{-24}{18} = -\frac{4}{3}\].

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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 for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula is: \[x = \frac{-b \-\ \sqrt{b^2 - 4ac}}{2a}\].
Here, the letters \('a', 'b', 'c'\) are the coefficients of the quadratic equation. The quadratic formula allows us to find the solutions (or roots) of the equation, even when factoring isn't possible or easy.
  • \(a\): Coefficient of \(x^2\)
  • \(b\): Coefficient of \(x\)
  • \(c\): Constant term
discriminant
The discriminant \(D = b^2 - 4ac\) is key in determining the nature of the roots of a quadratic equation.
It's the part under the square root in the quadratic formula: \[\sqrt{b^2 - 4ac}\].
Here is what the value of the discriminant tells us about the quadratic equation:
  • If \(D > 0\): The equation has two distinct real roots.
  • If \(D = 0\): The equation has exactly one real root, known as a repeated root.
  • If \(D < 0\): The equation has two complex roots (no real solutions).
repeated roots
Repeated roots occur when the discriminant is zero \(D = 0\). This means the quadratic equation has one unique solution that repeats itself.
When solving, you use the simplified quadratic formula for repeated roots: \[ x = \frac{-b}{2a} \].
For the problem \(9x^2 + 24x + 16 = 0\), we find: \[ x = \frac{-24}{2(9)} = \frac{-24}{18} = -\frac{4}{3} \].
coefficients
Coefficients are the numerical values in front of the variables in a polynomial. In the quadratic equation \(ax^2 + bx + c = 0\), \('a', 'b', 'c'\) are the coefficients:
  • \(a\): The coefficient of \(x^2\) determines the parabola's opening direction and width.
  • \(b\): The coefficient of \(x\) affects the symmetry and direction of the parabola.
  • \(c\): The constant term gives the y-intercept where the parabola crosses the y-axis.
For the equation given, \(9x^2 + 24x + 16 = 0\), the coefficients are \(a = 9, b = 24, c = 16\).

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