Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 42
Solve by (a) Completing the square (b) Using the quadratic formula $$ 4 x^{2}+4 x+1=0 $$
Short Answer
Expert verified
Question: Solve the quadratic equation \(4x^2 + 4x + 1 = 0\) using (a) completing the square, and (b) the quadratic formula.
Answer: The solutions for the given quadratic equation are \(x = 0\) and \(x = \frac{-1}{2}\).
Step by step solution
01
Rewrite the equation in the standard form
Rewrite the given equation as \(4(x^2 + x) + 1 = 0\).
02
Divide the equation by the leading coefficient
Divide the equation by the leading coefficient, which is 4, to make the leading term \(x^2\). We get \((x^2 + x) +\frac{1}{4}=0.\)
03
Add and subtract the square of half the coefficient of x
Take half of the coefficient of x (which is 1) and add its square (which is \(\frac{1}{4}\)) to the equation. Then, subtract \(\frac{1}{4}\) to keep the equation balanced. We get \((x^2 + x + \frac{1}{4}) - \frac{1}{4} + \frac{1}{4} = 0.\)
04
Simplify and use the square root property
The left-hand side now becomes a perfect square trinomial: \((x + \frac{1}{2})^2 = \frac{1}{4}\). Now, we can apply the square root property by taking the square root of both sides: $$x + \frac{1}{2} = \pm \frac{1}{2}.$$
05
Solve for x
Subtract \(\frac{1}{2}\) from both sides to solve for x: $$x = \frac{1}{2}(\pm 1 - 1).$$ This gives us two solutions: $$x = 0, \frac{-1}{2}.$$
Method (b) - Using the quadratic formula
06
Identify the coefficients
Identify the coefficients a, b, and c in the given equation: $$a = 4, b = 4, c = 1.$$
07
Plug the coefficients into the quadratic formula
Apply the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$ Substitute the values of a, b, and c into the formula: $$x = \frac{-(4) \pm \sqrt{(4)^2 - 4(4)(1)}}{2(4)}.$$
08
Simplify and solve for x
Simplify the expression: $$x = \frac{-4 \pm \sqrt{16 - 16}}{8}.$$ Notice that the expression under the square root is zero. Therefore, $$x = \frac{-4 \pm 0}{8}.$$ This gives us the same two solutions as in the completing the square method: $$x = 0, \frac{-1}{2}.$$
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
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 by transforming them into a perfect square trinomial. This method is particularly useful when you need exact solutions or to convert a quadratic equation into its vertex form. Here's how to do it:
- Start with a quadratic equation in the form of ax虏 + bx + c = 0.
- Divide every term by 'a', the coefficient of x虏, to simplify the leading term to x虏.
- Then, take half of the coefficient of x, square it, and add and subtract this value inside the equation to maintain balance.
- Now, rewrite the quadratic part as \( (x + d)^2 \), where d is half the coefficient of x.
- This transformation turns your equation into a square of a binomial, which you can solve by taking the square root of both sides.
Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation of the form ax虏 + bx + c = 0. It allows you to find the roots directly using the coefficients a, b, and c. To use the quadratic formula effectively, follow these steps:
- Identify the coefficients a, b, and c from your quadratic equation.
- Plug the coefficients into the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
- This formula calculates the solutions by using both the plus and minus configurations of the square root, allowing for potentially two different solutions depending on the discriminant \( b^2 - 4ac \).
- Simplify the calculations under the square root carefully.
Solving Equations
Solving quadratic equations involves finding the values of x that satisfy the equation ax虏 + bx + c = 0. The two primary methods taught in school are completing the square and the quadratic formula.
- For simple quadratics or when using vertex form, completing the square is an excellent method for deriving integer or simple rational solutions.
- The quadratic formula, however, is a powerful tool because it works universally on any quadratic equation, regardless of the discriminant's value or the complexity of solutions.
- With both methods, carefully balance and check your calculations, as precision is key to solving accurately.
- Once the solutions are found, you can verify by back-substituting into the original equation to ensure they satisfy it.
