Question
How do you solve a quadratic equation?
Answers
GPT-4o mini
To solve a quadratic equation, which is typically in the form \( ax^2 + bx + c = 0 \) (where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \)), you can use several methods:
1. **Factoring**: If the quadratic can be factored, express it as a product of two binomials. For example, if \( x^2 + 5x + 6 = 0 \) can be factored to \( (x + 2)(x + 3) = 0 \). Then set each factor to zero:
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
The solutions are \( x = -2 \) and \( x = -3 \).
2. **Completing the Square**: Rearrange the equation to isolate the \( x \) terms and then add or subtract values to complete the square. For example, for \( x^2 + 6x + 5 = 0 \):
\[
x^2 + 6x = -5
\]
Add \( 9 \) (which is \( (6/2)^2 \)) to both sides:
\[
(x + 3)^2 = 4
\]
Taking the square root of both sides gives:
\[
x + 3 = 2 \quad \Rightarrow \quad x = -1
\]
\[
x + 3 = -2 \quad \Rightarrow \quad x = -5
\]
The solutions are \( x = -1 \) and \( x = -5 \).
3. **Using the Quadratic Formula**: This method works for any quadratic equation. The formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here \( b^2 - 4ac \) is called the discriminant. It tells you about the nature of the roots:
- If \( > 0 \), there are two real and distinct solutions.
- If \( = 0 \), there is one real solution (a repeated root).
- If \( < 0 \), there are no real solutions (the roots are complex).
For example, for \( x^2 + 5x + 6 = 0 \):
\[
a = 1, \quad b = 5, \quad c = 6
\]
Calculate the discriminant:
\[
b^2 - 4ac = 5^2 - 4 \times 1 \times 6 = 25 - 24 = 1
\]
Plugging into the formula:
\[
x = \frac{-5 \pm \sqrt{1}}{2 \times 1} = \frac{-5 \pm 1}{2}
\]
This gives:
\[
x = \frac{-4}{2} = -2 \quad \text{and} \quad x = \frac{-6}{2} = -3
\]
The solutions are \( x = -2 \) and \( x = -3 \).
Choose any of these methods based on the specific quadratic equation and your preference!
1. **Factoring**: If the quadratic can be factored, express it as a product of two binomials. For example, if \( x^2 + 5x + 6 = 0 \) can be factored to \( (x + 2)(x + 3) = 0 \). Then set each factor to zero:
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
The solutions are \( x = -2 \) and \( x = -3 \).
2. **Completing the Square**: Rearrange the equation to isolate the \( x \) terms and then add or subtract values to complete the square. For example, for \( x^2 + 6x + 5 = 0 \):
\[
x^2 + 6x = -5
\]
Add \( 9 \) (which is \( (6/2)^2 \)) to both sides:
\[
(x + 3)^2 = 4
\]
Taking the square root of both sides gives:
\[
x + 3 = 2 \quad \Rightarrow \quad x = -1
\]
\[
x + 3 = -2 \quad \Rightarrow \quad x = -5
\]
The solutions are \( x = -1 \) and \( x = -5 \).
3. **Using the Quadratic Formula**: This method works for any quadratic equation. The formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here \( b^2 - 4ac \) is called the discriminant. It tells you about the nature of the roots:
- If \( > 0 \), there are two real and distinct solutions.
- If \( = 0 \), there is one real solution (a repeated root).
- If \( < 0 \), there are no real solutions (the roots are complex).
For example, for \( x^2 + 5x + 6 = 0 \):
\[
a = 1, \quad b = 5, \quad c = 6
\]
Calculate the discriminant:
\[
b^2 - 4ac = 5^2 - 4 \times 1 \times 6 = 25 - 24 = 1
\]
Plugging into the formula:
\[
x = \frac{-5 \pm \sqrt{1}}{2 \times 1} = \frac{-5 \pm 1}{2}
\]
This gives:
\[
x = \frac{-4}{2} = -2 \quad \text{and} \quad x = \frac{-6}{2} = -3
\]
The solutions are \( x = -2 \) and \( x = -3 \).
Choose any of these methods based on the specific quadratic equation and your preference!