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Chapter 7: Problem 39

Solve the given problems. All numbers are accurate to at least two significant digits. Find \(k\) if the equation \(x^{2}+4 x+k=0\) has a real double root.

Short Answer

Expert verified
The value of \(k\) is 4.

Step by step solution

01

Understand Double Root Condition

For an equation to have a double root, the discriminant must be zero. The discriminant of the quadratic equation \(ax^2 + bx + c = 0\) is \(b^2 - 4ac\). If this value is zero, the equation has a real double root.
02

Identify Coefficients

In the equation \(x^2 + 4x + k = 0\), identify the coefficients: \(a = 1\), \(b = 4\), and \(c = k\).
03

Apply Double Root Condition

Using the double root condition \(b^2 - 4ac = 0\), substitute the values \(a = 1\), \(b = 4\), and \(c = k\). This gives \(4^2 - 4 \times 1 \times k = 0\).
04

Solve the Equation

Simplify and solve the equation: \(16 - 4k = 0\). Isolating \(k\), we get \(4k = 16\). Divide both sides by 4 to find \(k\), resulting in \(k = 4\).

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Key Concepts

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

Real Double Root
A real double root occurs in a quadratic equation when the equation's graph touches the x-axis at exactly one point. To put it another way, the quadratic equation has a perfect square factor.
  • This happens specifically when both roots are equal, i.e., a repeated root.
  • Graphically, the parabola representing the quadratic equation just barely "kisses" the x-axis.
Finding a real double root for a specific example involves certain conditions. The most crucial of these is that the discriminant should be zero.
Discriminant
The discriminant is a vital component in determining the nature of the roots of a quadratic equation. For a quadratic equation of the form \(ax^2 + bx + c = 0\)the discriminant \(D\) is given by the formula: \[D = b^2 - 4ac\]
  • If \(D > 0\), the quadratic has two distinct real roots.
  • If \(D = 0\), the quadratic has one real double root.
  • If \(D < 0\), the quadratic has two complex (non-real) roots.
In the exercise, setting \(D = 0\) was necessary to ensure a real double root, meaning the solution revolves around ensuring this balance is achieved.
Polynomial Coefficients
In the context of quadratic equations, understanding polynomial coefficients is essential. Coefficients in a quadratic equation \(ax^2 + bx + c = 0\)

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Most popular questions from this chapter

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