Question
how to solve for a quadratic formula
Answers
Answer
simple terms, im stupid
Answer
simpler
Answered by
GPT-4o mini
The quadratic formula is used to find the solutions (or roots) of a quadratic equation, which is any equation that can be written in the standard form:
\[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here’s how to use the formula step by step:
1. **Identify the coefficients**:
Determine the values of \( a \), \( b \), and \( c \) from your quadratic equation.
2. **Calculate the discriminant**:
Compute the value of the discriminant, which is the expression under the square root:
\[
D = b^2 - 4ac
\]
The value of \( D \) will give you information about the nature of the roots:
- If \( D > 0 \): There are two distinct real roots.
- If \( D = 0 \): There is exactly one real root (a repeated root).
- If \( D < 0 \): There are no real roots (the roots are complex).
3. **Apply the quadratic formula**:
Substitute \( a \), \( b \), and \( D \) into the quadratic formula to find the values of \( x \):
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
4. **Simplify**:
Depending on the value of the discriminant, calculate the two potential values for \( x \) (using \( + \) and \( - \) in the formula).
### Example:
Consider the equation \( 2x^2 + 3x - 5 = 0 \).
1. Identify the coefficients: \( a = 2 \), \( b = 3 \), \( c = -5 \).
2. Calculate the discriminant:
\[
D = 3^2 - 4(2)(-5) = 9 + 40 = 49
\]
3. Apply the quadratic formula:
\[
x = \frac{-3 \pm \sqrt{49}}{2 \cdot 2} = \frac{-3 \pm 7}{4}
\]
4. Simplify:
- For \( x = \frac{-3 + 7}{4} = \frac{4}{4} = 1 \)
- For \( x = \frac{-3 - 7}{4} = \frac{-10}{4} = -2.5 \)
So the solutions to the quadratic equation \( 2x^2 + 3x - 5 = 0 \) are \( x = 1 \) and \( x = -2.5 \).
\[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here’s how to use the formula step by step:
1. **Identify the coefficients**:
Determine the values of \( a \), \( b \), and \( c \) from your quadratic equation.
2. **Calculate the discriminant**:
Compute the value of the discriminant, which is the expression under the square root:
\[
D = b^2 - 4ac
\]
The value of \( D \) will give you information about the nature of the roots:
- If \( D > 0 \): There are two distinct real roots.
- If \( D = 0 \): There is exactly one real root (a repeated root).
- If \( D < 0 \): There are no real roots (the roots are complex).
3. **Apply the quadratic formula**:
Substitute \( a \), \( b \), and \( D \) into the quadratic formula to find the values of \( x \):
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
4. **Simplify**:
Depending on the value of the discriminant, calculate the two potential values for \( x \) (using \( + \) and \( - \) in the formula).
### Example:
Consider the equation \( 2x^2 + 3x - 5 = 0 \).
1. Identify the coefficients: \( a = 2 \), \( b = 3 \), \( c = -5 \).
2. Calculate the discriminant:
\[
D = 3^2 - 4(2)(-5) = 9 + 40 = 49
\]
3. Apply the quadratic formula:
\[
x = \frac{-3 \pm \sqrt{49}}{2 \cdot 2} = \frac{-3 \pm 7}{4}
\]
4. Simplify:
- For \( x = \frac{-3 + 7}{4} = \frac{4}{4} = 1 \)
- For \( x = \frac{-3 - 7}{4} = \frac{-10}{4} = -2.5 \)
So the solutions to the quadratic equation \( 2x^2 + 3x - 5 = 0 \) are \( x = 1 \) and \( x = -2.5 \).
Answered by
GPT-4o mini
No problem! Let’s break it down into simpler steps.
### Steps to Solve a Quadratic Equation Using the Quadratic Formula
1. **What is a Quadratic Equation?**
It looks like this: \( ax^2 + bx + c = 0 \).
- \( a \), \( b \), and \( c \) are numbers. \( a \) should not be zero.
2. **Identify \( a \), \( b \), and \( c \)**
From your equation, find which numbers are \( a \), \( b \), and \( c \).
3. **Use the Quadratic Formula**
The formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
- **Don’t worry about the symbols yet**. Just follow these steps.
4. **Calculate the Discriminant**
- First, calculate \( b^2 - 4ac \).
- It helps you know what kind of answers you will get.
- If this number (let’s call it \( D \)) is:
- **Positive**: You’ll get 2 answers (two real solutions).
- **Zero**: You’ll get 1 answer (one real solution).
- **Negative**: You’ll get 2 complex answers (no real solutions).
5. **Plug Numbers into the Formula**
- Substitute \( a \), \( b \), and \( D \) back into the formula.
- Simplify to find \( x \).
### Example:
Let’s say we have the equation \( 2x^2 + 3x - 5 = 0 \).
1. **Identify \( a \), \( b \), and \( c \)**:
- \( a = 2 \)
- \( b = 3 \)
- \( c = -5 \)
2. **Calculate \( D \)**:
\[
D = b^2 - 4ac = 3^2 - 4(2)(-5) = 9 + 40 = 49
\]
(Here, \( D = 49 \), which is positive.)
3. **Use the Formula**:
\[
x = \frac{-b \pm \sqrt{D}}{2a} \\
x = \frac{-3 \pm \sqrt{49}}{2 \cdot 2} \\
x = \frac{-3 \pm 7}{4}
\]
4. **Find the Two Solutions**:
- **First solution**:
\[
x = \frac{-3 + 7}{4} = \frac{4}{4} = 1
\]
- **Second solution**:
\[
x = \frac{-3 - 7}{4} = \frac{-10}{4} = -2.5
\]
### Final Answers:
The solutions for \( 2x^2 + 3x - 5 = 0 \) are:
- \( x = 1 \)
- \( x = -2.5 \)
Feel free to ask if you have questions about any specific step!
### Steps to Solve a Quadratic Equation Using the Quadratic Formula
1. **What is a Quadratic Equation?**
It looks like this: \( ax^2 + bx + c = 0 \).
- \( a \), \( b \), and \( c \) are numbers. \( a \) should not be zero.
2. **Identify \( a \), \( b \), and \( c \)**
From your equation, find which numbers are \( a \), \( b \), and \( c \).
3. **Use the Quadratic Formula**
The formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
- **Don’t worry about the symbols yet**. Just follow these steps.
4. **Calculate the Discriminant**
- First, calculate \( b^2 - 4ac \).
- It helps you know what kind of answers you will get.
- If this number (let’s call it \( D \)) is:
- **Positive**: You’ll get 2 answers (two real solutions).
- **Zero**: You’ll get 1 answer (one real solution).
- **Negative**: You’ll get 2 complex answers (no real solutions).
5. **Plug Numbers into the Formula**
- Substitute \( a \), \( b \), and \( D \) back into the formula.
- Simplify to find \( x \).
### Example:
Let’s say we have the equation \( 2x^2 + 3x - 5 = 0 \).
1. **Identify \( a \), \( b \), and \( c \)**:
- \( a = 2 \)
- \( b = 3 \)
- \( c = -5 \)
2. **Calculate \( D \)**:
\[
D = b^2 - 4ac = 3^2 - 4(2)(-5) = 9 + 40 = 49
\]
(Here, \( D = 49 \), which is positive.)
3. **Use the Formula**:
\[
x = \frac{-b \pm \sqrt{D}}{2a} \\
x = \frac{-3 \pm \sqrt{49}}{2 \cdot 2} \\
x = \frac{-3 \pm 7}{4}
\]
4. **Find the Two Solutions**:
- **First solution**:
\[
x = \frac{-3 + 7}{4} = \frac{4}{4} = 1
\]
- **Second solution**:
\[
x = \frac{-3 - 7}{4} = \frac{-10}{4} = -2.5
\]
### Final Answers:
The solutions for \( 2x^2 + 3x - 5 = 0 \) are:
- \( x = 1 \)
- \( x = -2.5 \)
Feel free to ask if you have questions about any specific step!
Answered by
GPT-4o mini
Of course! Let’s make it really simple, step-by-step, using a simple example.
### Steps to Solve a Quadratic Equation
**Quadratic Equation Format:**
- Looks like this: **\( ax^2 + bx + c = 0 \)**
- Example: **\( 2x^2 + 3x - 5 = 0 \)**
### Simple Steps:
1. **Identify the Numbers:**
- Find **\( a \)**, **\( b \)**, and **\( c \)** in your equation.
- In our example:
- \( a = 2 \)
- \( b = 3 \)
- \( c = -5 \)
2. **Use the Formula:**
- The formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
3. **Calculate Inside the Square Root: (Discriminant)**
- Calculate \( b^2 - 4ac \):
- For our example:
\[
3^2 - 4 \cdot 2 \cdot (-5)
\]
\[
= 9 + 40 = 49
\]
- Now we see \( 49 \).
4. **Plug Numbers Back Into the Formula:**
- Now put everything into the formula:
\[
x = \frac{-3 \pm \sqrt{49}}{2 \cdot 2}
\]
5. **Solve for \( x \):**
- First, calculate the square root:
\[
\sqrt{49} = 7
\]
- This gives:
\[
x = \frac{-3 + 7}{4} \quad \text{and} \quad x = \frac{-3 - 7}{4}
\]
- So now calculate:
- **First solution:**
\[
x = \frac{-3 + 7}{4} = \frac{4}{4} = 1
\]
- **Second solution:**
\[
x = \frac{-3 - 7}{4} = \frac{-10}{4} = -2.5
\]
### Final Answers:
- The solutions for \( 2x^2 + 3x - 5 = 0 \) are:
- **\( x = 1 \)**
- **\( x = -2.5 \)**
### Key Points:
- **Identify \( a \), \( b \), and \( c \).**
- **Use the formula.**
- **Calculate inside the square root.**
- **Find the answers.**
Let me know if you need any more help!
### Steps to Solve a Quadratic Equation
**Quadratic Equation Format:**
- Looks like this: **\( ax^2 + bx + c = 0 \)**
- Example: **\( 2x^2 + 3x - 5 = 0 \)**
### Simple Steps:
1. **Identify the Numbers:**
- Find **\( a \)**, **\( b \)**, and **\( c \)** in your equation.
- In our example:
- \( a = 2 \)
- \( b = 3 \)
- \( c = -5 \)
2. **Use the Formula:**
- The formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
3. **Calculate Inside the Square Root: (Discriminant)**
- Calculate \( b^2 - 4ac \):
- For our example:
\[
3^2 - 4 \cdot 2 \cdot (-5)
\]
\[
= 9 + 40 = 49
\]
- Now we see \( 49 \).
4. **Plug Numbers Back Into the Formula:**
- Now put everything into the formula:
\[
x = \frac{-3 \pm \sqrt{49}}{2 \cdot 2}
\]
5. **Solve for \( x \):**
- First, calculate the square root:
\[
\sqrt{49} = 7
\]
- This gives:
\[
x = \frac{-3 + 7}{4} \quad \text{and} \quad x = \frac{-3 - 7}{4}
\]
- So now calculate:
- **First solution:**
\[
x = \frac{-3 + 7}{4} = \frac{4}{4} = 1
\]
- **Second solution:**
\[
x = \frac{-3 - 7}{4} = \frac{-10}{4} = -2.5
\]
### Final Answers:
- The solutions for \( 2x^2 + 3x - 5 = 0 \) are:
- **\( x = 1 \)**
- **\( x = -2.5 \)**
### Key Points:
- **Identify \( a \), \( b \), and \( c \).**
- **Use the formula.**
- **Calculate inside the square root.**
- **Find the answers.**
Let me know if you need any more help!
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