Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 10
In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}+2 x=4 $$
Short Answer
Expert verified
The roots are \( x = -1 + \sqrt{5} \) and \( x = -1 - \sqrt{5} \).
Step by step solution
01
Write the Equation in Standard Form
The given equation is \( x^2 + 2x = 4 \). To use the quadratic formula, the equation must be in the standard form \( ax^2 + bx + c = 0 \). Subtract 4 from both sides to get \( x^2 + 2x - 4 = 0 \).
02
Identify Coefficients
In the equation \( x^2 + 2x - 4 = 0 \), identify the coefficients: \( a = 1 \), \( b = 2 \), and \( c = -4 \).
03
Write the Quadratic Formula
The quadratic formula to find the roots of \( ax^2 + bx + c = 0 \) is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
04
Substitute the Coefficients into the Formula
Substitute \( a = 1 \), \( b = 2 \), and \( c = -4 \) into the quadratic formula: \[ x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-4)}}{2(1)} \].
05
Simplify the Expression under the Square Root
Calculate the expression under the square root (the discriminant): \( b^2 - 4ac = 2^2 - 4(1)(-4) = 4 + 16 = 20 \). The expression becomes \[ x = \frac{-2 \pm \sqrt{20}}{2} \].
06
Simplify the Square Root
Simplify \( \sqrt{20} \). Recall that \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \). Substitute back into the expression: \[ x = \frac{-2 \pm 2\sqrt{5}}{2} \].
07
Simplify the Entire Expression
Divide each term in the numerator by 2: \[ x = \frac{-2}{2} \pm \frac{2\sqrt{5}}{2} \]. This simplifies to \[ x = -1 \pm \sqrt{5} \].
08
Write the Final Answer
The roots of the equation \( x^2 + 2x = 4 \) are \( x = -1 + \sqrt{5} \) and \( x = -1 - \sqrt{5} \).
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.
Standard Form
When working with quadratic equations, it's important to get the equation into a special format called Standard Form. This form is always written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are numbers called coefficients.
To transform any quadratic equation into Standard Form, you might need to do some rearranging. For example, if you start with an equation like \( x^2 + 2x = 4 \), first move all terms to one side of the equation so that the other side equals zero. This means subtracting 4 from both sides:
Now, the equation becomes \( x^2 + 2x - 4 = 0 \), which is in Standard Form.
To transform any quadratic equation into Standard Form, you might need to do some rearranging. For example, if you start with an equation like \( x^2 + 2x = 4 \), first move all terms to one side of the equation so that the other side equals zero. This means subtracting 4 from both sides:
- Subtract 4 from both sides of the equation
Now, the equation becomes \( x^2 + 2x - 4 = 0 \), which is in Standard Form.
Coefficients
In the context of the quadratic equation in Standard Form \( ax^2 + bx + c = 0 \), coefficients are the numbers that multiply the variables. Understanding these coefficients is key to using the quadratic formula.
Let's break down their roles:
For example, in the equation \( x^2 + 2x - 4 = 0 \):
Let's break down their roles:
- \(a\): This is the coefficient of \(x^2\). It's the number in front of the \(x^2\) term.
- \(b\): This is the coefficient of \(x\). It鈥檚 the number attached to the \(x\) term.
- \(c\): This is the constant term; it has no variable attached to it.
For example, in the equation \( x^2 + 2x - 4 = 0 \):
- \(a = 1\), because it鈥檚 \(1\) times \(x^2\).
- \(b = 2\), because it鈥檚 \(2\) times \(x\).
- \(c = -4\), since it鈥檚 the constant, just \(-4\).
Discriminant
The discriminant is part of the quadratic formula and is found in the expression under the square root sign: \(b^2 - 4ac\). It gives valuable information about the nature of the roots of the quadratic equation.
Here's how to interpret the discriminant:
For the equation \(x^2 + 2x - 4 = 0\), the discriminant is calculated as follows:
Here's how to interpret the discriminant:
- If \(b^2 - 4ac > 0\), there are two distinct real roots.
- If \(b^2 - 4ac = 0\), there is exactly one real root (also known as a repeated root).
- If \(b^2 - 4ac < 0\), there are no real roots, only complex roots.
For the equation \(x^2 + 2x - 4 = 0\), the discriminant is calculated as follows:
- Using \(a = 1\), \(b = 2\), \(c = -4\)
- \(b^2 - 4ac = 2^2 - 4(1)(-4) = 4 + 16 = 20\)
Square Root Simplification
Simplifying square roots is crucial in solving quadratic equations using the quadratic formula, especially when dealing with irrational numbers. After calculating the discriminant \(b^2 - 4ac\), resulting in a number like 20, we need to simplify \(\sqrt{20}\).
Here's the process for simplifying square roots:
Applying this to the quadratic formula solution, \(x = \frac{-2 \pm \sqrt{20}}{2}\) becomes \(x = \frac{-2 \pm 2\sqrt{5}}{2}\), and further simplifies to \(x = -1 \pm \sqrt{5}\). This simplification step is key for expressing the roots in their simplest radical form.
Here's the process for simplifying square roots:
- Check if the number under the square root can be broken down into perfect square factors. For 20, it can be expressed as \(4 \times 5\).
- The square root of a product is equal to the product of the square roots: \(\sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5}\).
- We know that \(\sqrt{4} = 2\), so \(\sqrt{20} = 2\sqrt{5}\).
Applying this to the quadratic formula solution, \(x = \frac{-2 \pm \sqrt{20}}{2}\) becomes \(x = \frac{-2 \pm 2\sqrt{5}}{2}\), and further simplifies to \(x = -1 \pm \sqrt{5}\). This simplification step is key for expressing the roots in their simplest radical form.
