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Chapter 11: Problem 15

What are the steps used to solve a quadratic equation by completing the square?

Short Answer

Expert verified
To solve a quadratic equation by completing the square, follow these steps: (1) Write the equation in standard form \(ax^2 + bx + c = 0\). (2) Move the constant term to the right side. (3) Complete the square by adding \((b/2)^2\) to both sides, and rewrite the left side as a binomial squared. (4) Solve for the variable by taking the square root of both sides and considering positive and negative square root cases. For example, given the equation \(x^2 - 6x + 8 = 0\), the solutions are \(x = 4\) and \(x = 2\).

Step by step solution

01

Write the Quadratic Equation in Standard Form

The standard form of a quadratic equation is given by \(ax^2 + bx + c = 0\), where a, b, and c are constants. For example, let's consider the quadratic equation \(x^2 - 6x + 8 = 0\).
02

Move the Constant Term to the Right Side

Subtract the constant term (c) from both sides of the equation. \(x^2 - 6x + 8 - 8 = 0 - 8\) \(x^2 - 6x = -8\)
03

Complete the Square on the Left Side

To complete the square, add the square of half of the coefficient of the x term (b) to both sides of the equation. The formula to find the square of half the coefficient is: \((b / 2)^2\). \(((-6) / 2)^2 = ( -3)^2 = 9\) So, we add 9 to both sides of the equation: \(x^2 - 6x + 9 = -8 + 9\) \(x^2 - 6x + 9 = 1\) Now, the left side becomes a perfect square trinomial, and it can be written as a binomial squared: \((x - 3)^2 = 1\)
04

Solve for x

To solve for x, we take the square root of both sides: \(卤\sqrt{(x - 3)^2} = 卤\sqrt{1} \) \(卤(x - 3) = 卤1 \) Now, solve for x in two cases: Case 1: Positive square root \(x - 3 = 1\) \(x = 1 + 3\) \(x = 4\) Case 2: Negative square root \(x - 3 = -1\) \(x = -1 + 3\) \(x = 2\) So, the solutions for the quadratic equation are \(x = 4\) and \(x = 2\).

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Key Concepts

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

Completing the Square
Completing the square is a method used to solve quadratic equations. It transforms a quadratic equation into a perfect square trinomial. This makes it easier to solve.
To complete the square, follow these steps:
  • Start with a quadratic equation in standard form: \( ax^2 + bx + c = 0 \).
  • Move the constant term \( c \) to the other side of the equation.
  • Divide the coefficient of \( x \) (which is \( b \)) by 2, then square the result. This is the key to completing the square.
  • Add this squared value to both sides of the equation, turning the left side into a perfect square trinomial.
  • Factor the left side into \((x + d)^2\) form, where \(d\) is half of the \(x\) coefficient, as obtained from an earlier step.
Once this transformation is complete, solve the equation by taking the square root of both sides. This method is particularly useful when factoring is not straightforward.
Quadratic Equations
Quadratic equations are polynomial equations of the second degree. They have the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).

These equations can graph as parabolas. The solutions for quadratic equations can be found using different methods, such as:
  • Factoring, when it is possible.
  • The Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • Completing the Square, which we discussed above.
Each method has its own advantages and is suited for different forms of quadratic equations. Completing the square is especially useful for learning about the structure of quadratics and connecting the algebraic form to the geometric representation.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating them. It is the language through which we describe relationships between quantities and includes both simple equations and complex structures like polynomials.

Quadratic equations, which we often solve using algebraic techniques like completing the square, are a vital part of this field. Algebra helps us:
  • Express and solve equations analytically.
  • Model real-world scenarios mathematically.
  • Understand patterns and structures in numbers and equations.
Learning algebra builds foundational skills necessary for more advanced mathematics, enabling students to handle a wide range of mathematical problems.

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Most popular questions from this chapter

Solve. An object is propelled upward from a height of \(4 \mathrm{ft}\). The height \(h\) of the object (in feet) \(t\) sec after the object is released is given by $$ h=-16 t^{2}+60 t+4 $$ a) How long does it take the object to reach a height of \(40 \mathrm{ft} ?\) b) How long does it take the object to hit the ground?

Write an equation and solve. The width of a rectangular piece of cardboard is 8 in. less than its length. A square piece that measures 2 in. on each side is cut from each corner, then the sides are turned up to make a box with volume 480 in \(^{3}\). Find the length and width of the original piece of cardboard.

Solve. \(\frac{2}{(c+6)^{2}}+\frac{2}{(c+6)}=1\)

Divide. Write the result in the form \(a+b i\). $\frac{2+3 i}{5-6 i}$$

Divide. Write the result in the form \(a+b i\). $\frac{9 i}{-4+10 i}$$
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