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Chapter 8: Problem 29

Solve. (Find all complex-number solutions.) \(7 x(x+2)+5=3 x(x+1)\)

Short Answer

Expert verified
The solutions are \(x = \frac{-11 + \sqrt{41}}{8}\) and \(x = \frac{-11 - \sqrt{41}}{8}\).

Step by step solution

01

- Expand the Equation

First, distribute the terms on both sides of the equation. Expand the left side: \[7x(x+2) = 7x^2 + 14x\] Expand the right side: \[3x(x+1) = 3x^2 + 3x\] Now, the equation becomes: \[7x^2 + 14x + 5 = 3x^2 + 3x\]
02

- Move All Terms to One Side

Subtract all the terms on the right side from both sides to set the equation to zero: \[7x^2 + 14x + 5 - 3x^2 - 3x = 0\] Simplify: \[4x^2 + 11x + 5 = 0\]
03

- Solve the Quadratic Equation

To solve the quadratic equation, use the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] In our equation, \[a = 4, b = 11, c = 5\] Substitute these values into the formula: \[x = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 4 \cdot 5}}{2 \cdot 4}\] This simplifies to: \[x = \frac{-11 \pm \sqrt{121 - 80}}{8}\] \[x = \frac{-11 \pm \sqrt{41}}{8}\]
04

- Simplify the Expressions

Now we have the two solutions for the equation: \[x_1 = \frac{-11 + \sqrt{41}}{8}\] \[x_2 = \frac{-11 - \sqrt{41}}{8}\] Since the discriminant is positive (\(41\)), both solutions are real numbers.

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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 mathematical tool used to solve quadratic equations. A quadratic equation is typically in the form \[ax^2 + bx + c = 0\], where \(a, b,\) and \(c\) are constants. To find the roots (solutions) of a quadratic equation, we use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. This formula provides the solutions for \(x\) based on the values of \(a, b,\) and \(c\).
  • \(a\): Coefficient of the \(x^2\) term
  • \(b\): Coefficient of the \(x\) term
  • \(c\): Constant term
Substituting the constants into this formula allows you to find both solutions for the quadratic equation. In our example, substituting \(a = 4, b = 11,\) and \(c = 5\) into the quadratic formula provides the solutions for \(x\). Each root is calculated using the \( \pm \sqrt{} \) notation, giving us two possible values for \(x\). This guarantees you find both the possible roots of the equation.
Discriminant
The discriminant is a component of the quadratic formula within the square root symbol: \(b^2 - 4ac\). It determines the nature and number of solutions for a quadratic equation. Depending on the value of the discriminant, you can have:
  • A positive discriminant (\(\Delta > 0\)): Two distinct real solutions
  • A discriminant of zero (\(\Delta = 0\)): One real solution (a repeated root)
  • A negative discriminant (\(\Delta < 0\)): Two complex solutions (no real solutions)
In our example problem, \(b^2 - 4ac = 121 - 80 = 41\), which is positive. Therefore, the quadratic equation \(4x^2 + 11x + 5 = 0\) has two distinct real number solutions.
Expanding Equations
Expanding equations involves distributing and simplifying terms to convert the original equation into a polynomial form. This helps in setting up the equation to use the quadratic formula effectively.For our example, the original problem is \(7x(x+2)+5=3x(x+1)\). By expanding, we distribute the constants and simplify the terms on both sides of the equation. This process looks like:
  • Expanding the left side: \(7x(x+2) = 7x^2 + 14x\)
  • Expanding the right side: \(3x(x+1) = 3x^2 + 3x\)
Bringing all terms to one side gives us \(4x^2 + 11x + 5 = 0\), a standard quadratic equation format. Once the equation is in this format, it is easier to apply the quadratic formula to find the solutions for \(x\).

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