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Chapter 9: Problem 67

Explain how to solve $$x^{2}+6 x+8=0$$ using the quadratic formula,

Short Answer

Expert verified
The solutions to the equation \(x^{2}+6 x+8=0\) are \(x = -2\) and \(x = -4\).

Step by step solution

01

Identify the coefficients

From the equation \(x^{2}+6 x+8=0\), we can identify the coefficients as follows: \(a = 1\), \(b = 6\) and \(c = 8\).
02

Substitute the coefficients into the quadratic formula

Substitute \(a = 1\), \(b = 6\) and \(c = 8\) into the quadratic formula, yielding \[x = \frac{-6 \pm \sqrt{(6)^{2} - 4*(1)*(8)}}{2*(1)}\].
03

Simplify the equation

Simplify the equation to get \[x = \frac{-6 \pm \sqrt{36 - 32}}{2}\].
04

Calculate the square root and simplify further

After calculating the square root, the equation simplifies to: \[x = \frac{-6 \pm 2}{2}\].
05

Finally, calculate x

The equation thus yields two solutions: \(x = \frac{-6 + 2}{2} = -2\) and \(x = \frac{-6 - 2}{2} = -4\).

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Key Concepts

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

Quadratic Equation
A quadratic equation is a polynomial equation of the second degree. This means it has the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(aeq0\).
Quadratic equations are essential in various fields like physics, engineering, and economics due to their application in calculating different curves and parabolic trajectories. Here, the equation \(x^2 + 6x + 8 = 0\) is a typical quadratic equation.
Coefficients
In the context of a quadratic equation, the coefficients are the numbers \(a\), \(b\), and \(c\) in the equation \(ax^2 + bx + c = 0\).
They determine the shape and position of the parabola that represents the quadratic equation.
  • \(a\): The coefficient of \(x^2\). It indicates the direction and width of the parabola. Here, \(a = 1\).
  • \(b\): The coefficient of \(x\). This affects the symmetry of the parabola. In our equation, \(b = 6\).
  • \(c\): The constant term. It impacts the vertical position of the parabola. For this equation, \(c = 8\).
Recognizing these coefficients is the first step in solving quadratic equations using methods like the quadratic formula.
Solving Equations
Solving equations is a fundamental concept in mathematics, crucial for determining unknown values in algebra.
For quadratic equations, one of the most effective methods is using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
This formula is versatile and can solve any quadratic equation, but it requires identifying the correct coefficients.
By substituting the coefficients \(a = 1\), \(b = 6\), and \(c = 8\) into the formula, we simplify and calculate to find both solutions for \(x\).
The solutions \(-2\) and \(-4\) are then derived, demonstrating the equation's potential solutions.
Mathematics Education
Understanding how to solve quadratic equations is a vital component of mathematics education.
This skill lays the groundwork for more advanced mathematical topics and problem-solving techniques.
Learning different methods to solve quadratic equations, like factoring, completing the square, and using the quadratic formula, enhances problem-solving skills.
The quadratic formula, in particular, is powerful as it works universally for any quadratic equation, provided the calculations are correct.
  • It encourages analytical thinking as students learn to identify patterns and strategies applicable to diverse mathematical problems.
  • Practicing these methods improves numerical fluency and comprehension, essential skills in both academic and real-world contexts.
Engaging with quadratic equations fosters a deeper appreciation of mathematical concepts and their applications.

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Most popular questions from this chapter

The hypotenuse of a right triangle is 6 feet long. One leg is 1 foot shorter than the other. Find the lengths of the legs. Round to the nearest tenth of a foot.

The weight of a human fetus is modeled by the formula \(W=3 t^{2}\) where \(W\) is the weight, in grams, and \(t\) is the time, in weeks, with \(0 \leq t \leq 39 .\) Use this formula to solve Exercises \(73-74\) After how many weeks does the fetus weigh 192 grams?

Your friend heard that functions are studied in intermediate and college algebra courses. He asks you what a function is. Provide him with a clear, relatively concise response.

a. Use a graphing utility to graph \(y=2 x^{2}-82 x+720\) in a standard viewing rectangle. What do you observe? b. Find the coordinates of the vertex for the given quadratic equation. c. The answer to part (b) is \((20.5,-120.5) .\) Because the leading coefficient, 2 of \(y=2 x^{2}-82 x+720\) is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at \(x=20.5,\) the setting for \(x\) should extend past this, so try Xmin \(=0\) and \(\mathrm{Xmax}=30 .\) The setting for \(y\) should include (and probably go below) the \(y\) -coordinate of the graph's minimum point, so try Ymin \(=-130 .\) Experiment with Ymax until your utility shows the parabola's major features. d. In general, explain how knowing the coordinates of a parabola's vertex can help determine a reasonable viewing rectangle on a graphing utility for obtaining a complete picture of the parabola.

The square of the difference between a number and 7 is 16 Find the number \((s)\)
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