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Chapter 16: Problem 30

For Exercises 7 to \(47,\) solve by completing the square. $$x^{2}-x=1$$

Short Answer

Expert verified
The solution to the equation is \(x = 0.5 \pm \sqrt{1.25}\).

Step by step solution

01

Move Constants to Right Side

Rearrange the equation to move the constant term to the right side of the equation. For our equation, this step gives us: \(x^2 - x - 1 = 0\).
02

Complete the Square

Complete the square on the left side of the equation. The coefficient of \(x\) is -1. Half of -1 is -0.5, and \((-0.5)^2 = 0.25\). Add this number to both sides of the equation to form a perfect square on the left side. Thus, our equation now becomes: \((x^2 - x + 0.25) - 1 - 0.25 = 0\), simplifying this equation will give us: \((x - 0.5)^2 - 1.25 = 0\).
03

Solve for x

Now solve for \(x\). Add 1.25 to both sides of the equation and then take the square root of both sides, with consideration of \(\pm\) for the square root result. We then have \(x - 0.5 = \pm \sqrt{1.25}\). We add 0.5 to both sides to completely isolate \(x\), yielding the solution \(x = 0.5 \pm\sqrt{1.25}\).

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Key Concepts

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

Algebraic Equations
Algebraic equations are the backbone of algebra and represent the relationship between different variables and constants. An equation is a statement that two expressions are equal, and the solutions to the equations are the values that make the equation true.

To solve algebraic equations, we perform various operations such as addition, subtraction, multiplication, and division, while keeping the equation balanced. These operations should be done to both sides of the equation to maintain equality. For example, if we have the equation x + 3 = 5, we can subtract 3 from both sides to find x = 2.
Quadratic Equations
Quadratic Equations are a special type of algebraic equation that can be identified by their standard form, which is \( ax^2 + bx + c = 0 \) where \( a \) is not equal to 0. These equations must have a variable raised to the second power, which makes the graph of such functions a parabola.

To solve them, methods such as factoring, using the quadratic formula, graphing, or completing the square are employed. Completing the square is a powerful technique that transforms a quadratic equation into a perfect square trinomial, thus making it easier to solve. It involves finding a number that, when added to both sides of the equation, allows us to create a binomial square on one side of the equation.
Solving for x
Solving for \( x \) is a fundamental skill in algebra. It requires isolating the variable \( x \) on one side of the equation to determine its value. This process can be simple for linear equations, but for quadratic equations, it often requires more steps.

Completing the square is a technique especially useful when the quadratic equation cannot be easily factored. By manipulating the equation step by step鈥攕tarting by moving constants to one side, adding a specific value to both sides to form a perfect square trinomial, and then solving for \( x \)鈥攐ne can find the solutions to the equation. Always remember to consider both the positive and negative square roots when taking the square root of both sides, as square roots have two possible values.

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