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Chapter 9: Problem 13

Solve quadratic equation by completing the square. \(x^{2}+4 x=5\)

Short Answer

Expert verified
The solutions for the given quadratic equation are \(x=1\) and \(x=-5\).

Step by step solution

01

Rearrange the equation

The first step is to rearrange the equation to a form that can allow completing the square. Start with the equation \(x^{2}+4 x=5\). Move the constant from the left-hand side to the right-hand side of the equation. The rearranged equation is \(x^{2}+4x-5=0\).
02

Complete the square

Next, the term that complete the square on the left-hand side of the equation is \((\frac{4}{2})^{2} =4\). So, add 4 to both sides of the equation. The resulting equation is \(x^{2}+4x + 4=5 + 4\), which simplifies to \((x+2)^{2} = 9\).
03

Solve for \(x\)

Finally, take the square root of both sides of the equation to solve for \(x\). This yields two solutions because \(\sqrt{9}\) can be +3 or -3. Therefore, \(x+2 = +3\) or \(x+2 = -3\). Solving these two equations, you get \(x=1\) or \(x=-5\).

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Key Concepts

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

Quadratic Equations
Quadratic equations are polynomial equations of the second degree. These equations have the standard form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). Any equation involving at least one squared variable term is classified as a quadratic equation.
Quadratic equations are crucial in various fields such as physics, engineering, and finance. They model scenarios where variables have a squared relationship. The goal often is to find the value(s) of \(x\) that satisfy the quadratic equation.
There are multiple methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Each method can be beneficial in different scenarios depending on the equation's structure.
Rearranging Equations
Rearranging equations helps us to make them easier to solve by manipulating their form. For quadratic equations, rearranging often involves moving all terms to one side to set the equation to zero.
For instance, if we start with a quadratic equation such as \(x^2 + 4x = 5\), our first step is to move every numeric or non-variable term to one side: \[ x^2 + 4x - 5 = 0 \]
This gives us a standard form of the quadratic equation. Rearranging properly allows us to use other techniques, like completing the square or factoring, to solve the equation. Ensuring that the equation is correctly rearranged is essential for the success of any solving method.
Consider swapping signs when moving terms across the equation and check the work to prevent arithmetic errors.
Solving Equations Step-by-Step
Solving equations step-by-step is a systematic way to find solutions. It ensures that each part of the process is clear and any mistakes are easier to spot and correct.
For the quadratic equation \(x^2 + 4x - 5 = 0\):
  • Complete the Square: To solve by completing the square, you need to create a perfect square trinomial. Divide \(b\), the coefficient of \(x\), by 2 and square it. Here, \((\frac{4}{2})^2 = 4\). Add 4 to both sides.
  • Write as a Perfect Square: The left side transforms to \((x+2)^2\), giving \((x+2)^2 = 9\).
  • Take the Square Root: Apply the square root to both sides, remembering it's a positive and negative root. \(x+2 = 3\) or \(x+2 = -3\).
  • Solve: Solve for \(x\) to find the solution set: \(x = 1\) and \(x = -5\).
Following these steps carefully ensures clarity and success in solving the equation.

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Most popular questions from this chapter

Can squaring a real number result in a negative number? Based on your answer, are \(\sqrt{-1}\) and \(\sqrt{-4}\) real numbers?

For the quadratic equation \(-2 x^{2}+3 x=0,\) we have \(a=-2, b=3,\) and \(c=0\).

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Here are two sets of ordered pairs: $$\begin{array}{l} \text { set } 1:\\{(1,5),(2,5)\\} \\ \text { set } 2:\\{(5,1),(5,2)\\} \end{array}$$ In which set is each \(x\) -coordinate paired with one and only one \(y\) -coordinate?
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