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Chapter 7: Problem 20

Solve each equation by completing the square. $$ 2 x^{2}+4 x-1=0 $$

Short Answer

Expert verified
The solutions to the equation are:

Step by step solution

01

- Move the Constant Term

First, move the constant term to the other side of the equation.Starting with: Let鈥檚 add 1 to both sides of the equation: The equation becomes:
02

- Make the Coefficient of \(x^2\) Equal to 1

Divide every term in the equation by the coefficient of \(x^2\), which in this case is 2.The equation becomes:
03

- Complete the Square

Take half of the coefficient of \(x\), square it, and add it to both sides of the equation. The coefficient of \(x\) is 2, so half of it is 1, and squaring it givesThe equation becomes:
04

- Write the Left Side as a Binomial Square

The left-hand side of the equation can now be written as a square of a binomial. Rewrite it: So the equation now reads:
05

- Solve for the Variable

Finally, solve for \(x\) by taking the square root on both sides and then isolating \(x\).

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Key Concepts

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

solving quadratic equations
Solving quadratic equations is an important skill in algebra. Quadratic equations are any equations of the form \(ax^2 + bx + c = 0\). These equations are crucial because they model many real-world phenomena, from physics to finance.
One powerful method to solve them is completing the square. Completing the square turns the quadratic equation into a perfect square trinomial, making it easier to solve. This is especially useful when the quadratic equation doesn't factor easily.
Let's break down how we solved the given equation step-by-step.
algebraic manipulation
Algebraic manipulation involves rearranging equations to solve for a variable. This is essential for completing the square.
In our case, we needed to isolate the quadratic term. First, move the constant term to the other side by adding or subtracting it. Here, we added 1 to both sides:
\[ 2x^2 + 4x - 1 + 1 = 0 + 1 \]
which simplifies to:
\[ 2x^2 + 4x = 1 \]
Next, we made the coefficient of \(x^2\) equal to 1 by dividing every term by 2:
\[ x^2 + 2x = \frac{1}{2} \]
This simplification makes it easier to complete the square.
binomial square
A binomial square is a perfect square trinomial of the form \((x + d)^2\). To achieve this, we complete the square by adding a term to both sides of the equation.
Take half of the coefficient of \(x\), which is 2, so half is 1. Then, square it to get 1. Add this to both sides:
\[ x^2 + 2x + 1 = \frac{1}{2} + 1 \]
which simplifies to:
\[ x^2 + 2x + 1 = \frac{3}{2} \]
Now, the left side can be written as a binomial square:
\[ (x + 1)^2 = \frac{3}{2} \]
Solve for \(x\) by taking the square root of both sides:
\[ x + 1 = \frac{\text{卤}\root2 ({3}/{2})} \]
Finally, isolate \(x\):
\[ x = -1 \text{卤} \root2 ({3}/{2}) \]

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Most popular questions from this chapter

Solve each equation by factoring using integers, if possible. If an equation can't be solved in this way, explain why. $$ 14 e=e^{2}+24 $$

Solve each equation using the quadratic formula, if possible. $$ c^{2}-10=0 $$

Identify the values of \(a, b,\) and \(c\) in each equation by rearranging it into the form of the general quadratic equation, \(a x^{2}+b x+c=0\) $$ 7-w^{2}=w+2.5 w^{2} $$

Factor each quadratic expression that can be factored using integers. Identify those that cannot, and explain why they can't be factored. $$ h^{2}-3 h-28 $$

Find the number of solutions to each quadratic equation without actually solving the equation. Explain how you know your answers are correct. $$ 9 x^{2}+12 x+4=0 $$
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