Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 66
Solve the quadratic equation by completing the square. $$x^{2}+3 x=4$$
Short Answer
Expert verified
The solutions to the equation are \(x = 1\) and \(x = -4\).
Step by step solution
01
Re-arrange the Equation.
First, we need to re-arrange the equation so that we have all terms except the constant on one side and the constant on the other side.\nSo, we get \(x^{2}+3x - 4 = 0\)
02
Rewrite the Equation in the 'Completing the Square' Format
We can rewrite this equation as \((x + \frac{3}{2})^{2} - (\frac{3}{2})^{2} - 4 = 0 \), And if we simplify further, we get \((x + \frac{3}{2})^{2} - \frac{9}{4} - 4 = 0\)
03
Solve for x using Square root method
Next, we solve for x by first isolating the square term. To do this, we add the constants to both sides to get \((x + \frac{3}{2})^{2}= \frac{9}{4} + 4\), which simplifies to \((x + \frac{3}{2})^{2} = \frac{25}{4}\). Taking the square root of both sides give us \(x + \frac{3}{2} = \pm\frac{5}{2}\). Solving for x gives \(x = \frac{5}{2} - \frac{3}{2}\) or \(x = -\frac{5}{2} - \frac{3}{2}\). This simplifies to \(x = 1\) or \(x = -4\)
04
Checking the solution
If we put \(x = 1\) or \(x = -4\) back into the original equation \(x^{2}+3x - 4 = 0\), the equation is satisfied, so our solutions are correct.
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 technique used to solve quadratic equations by changing the equation into a perfect square trinomial. This method is especially useful for equations not easily factored. Here鈥檚 how it works with our example. Let's start with the quadratic equation: \[ x^{2} + 3x = 4 \]To complete the square, follow these steps:
- First, move the constant term to the other side of the equation. That gives us: \( x^{2} + 3x - 4 = 0 \).
- Next, focus on the quadratic and linear terms \( x^{2} + 3x \). We add and subtract a value that makes the expression a perfect square, which means finding half of the linear coefficient, squaring it, and adding it back to form a complete square trinomial. In this case, half of 3 is \( \frac{3}{2} \), and when squared, it is \( \frac{9}{4} \).
- Rewrite the equation with the complete square trinomial: \[ (x + \frac{3}{2})^{2} = \frac{9}{4} + 4 \]
Step-by-Step Algebra
Breaking down algebraic problems into small, manageable steps can make solving them much simpler. Here's how this step-by-step strategy applies to solving a quadratic equation by completing the square. Consider our rewritten equation: \[ (x + \frac{3}{2})^{2} - (\frac{9}{4}) - 4 = 0 \]Here's the step-by-step breakdown:
- Rearrange the equation so that all terms but the constant are on one side, and the constant is on the opposite side, yielding: \( x^{2} + 3x - 4 = 0 \).
- Factor the resulting perfect square trinomial: \[ (x + \frac{3}{2})^{2} - \frac{25}{4} = 0 \].
- Move the constant term on the left to the right-hand side: \[ (x + \frac{3}{2})^{2} = \frac{25}{4} \].
Solving Equations
Solving equations, particularly quadratics, involves isolating the variable to uncover its value. After completing the square for: \[ (x + \frac{3}{2})^{2} = \frac{25}{4} \]Here鈥檚 how you can solve it:
- Take the square root of both sides of the equation to get rid of the square: \[ x + \frac{3}{2} = \pm \frac{5}{2} \]. This means two cases: \( x + \frac{3}{2} = \frac{5}{2} \) and \( x + \frac{3}{2} = -\frac{5}{2} \).
- Subtract \( \frac{3}{2} \) from each side in both cases to solve for \( x \):
- For \( x + \frac{3}{2} = \frac{5}{2} \), subtracting \( \frac{3}{2} \) gives \( x = 1 \).
- For \( x + \frac{3}{2} = -\frac{5}{2} \), subtracting \( \frac{3}{2} \) results in \( x = -4 \).
- Verify by plugging each solution back into the original quadratic equation to ensure they satisfy it. Both solutions, \( x = 1 \) and \( x = -4 \), fit the equation \( x^{2} + 3x - 4 = 0 \), confirming their accuracy.
