Question
How do I solve a quadratic equation using the quadratic formula?
Answers
GPT-4o mini
To solve a quadratic equation using the quadratic formula, you can follow these steps:
1. **Identify the standard form of the quadratic equation**: A quadratic equation is typically written in the standard form:
\[
ax^2 + bx + c = 0
\]
where \( a \), \( b \), and \( c \) are coefficients and \( a \neq 0 \).
2. **Write down the quadratic formula**: The quadratic formula is given by:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\]
This formula provides the solutions for \( x \) (the roots of the quadratic equation).
3. **Calculate the discriminant**: The expression under the square root, \( b^2 - 4ac \), is called the discriminant. It determines the nature of the roots:
- If the discriminant is positive (\( > 0 \)), there are two distinct real solutions.
- If it is zero (\( = 0 \)), there is exactly one real solution (a repeated root).
- If it is negative (\( < 0 \)), there are no real solutions, but two complex solutions.
4. **Substitute the values of \( a \), \( b \), and \( c \) into the formula**: Plug the coefficients into the quadratic formula to calculate the values of \( x \).
5. **Simplify the expression**: Perform the calculations to find \( x \).
### Example
Let's solve the quadratic equation \( 2x^2 - 4x - 6 = 0 \).
1. Identify \( a \), \( b \), and \( c \):
\[
a = 2, \quad b = -4, \quad c = -6
\]
2. Write the quadratic formula:
\[
x = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4 \cdot 2 \cdot (-6)}}}}{{2 \cdot 2}}
\]
3. Calculate the discriminant:
\[
(-4)^2 - 4 \cdot 2 \cdot (-6) = 16 + 48 = 64
\]
4. Substitute and simplify:
\[
x = \frac{{4 \pm \sqrt{64}}}{{4}} = \frac{{4 \pm 8}}{{4}}
\]
This gives two solutions:
\[
x_1 = \frac{{4 + 8}}{{4}} = \frac{12}{4} = 3
\]
\[
x_2 = \frac{{4 - 8}}{{4}} = \frac{-4}{4} = -1
\]
5. Therefore, the solutions to the equation \( 2x^2 - 4x - 6 = 0 \) are:
\[
x = 3 \quad \text{and} \quad x = -1
\]
And that's how you solve a quadratic equation using the quadratic formula!
1. **Identify the standard form of the quadratic equation**: A quadratic equation is typically written in the standard form:
\[
ax^2 + bx + c = 0
\]
where \( a \), \( b \), and \( c \) are coefficients and \( a \neq 0 \).
2. **Write down the quadratic formula**: The quadratic formula is given by:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\]
This formula provides the solutions for \( x \) (the roots of the quadratic equation).
3. **Calculate the discriminant**: The expression under the square root, \( b^2 - 4ac \), is called the discriminant. It determines the nature of the roots:
- If the discriminant is positive (\( > 0 \)), there are two distinct real solutions.
- If it is zero (\( = 0 \)), there is exactly one real solution (a repeated root).
- If it is negative (\( < 0 \)), there are no real solutions, but two complex solutions.
4. **Substitute the values of \( a \), \( b \), and \( c \) into the formula**: Plug the coefficients into the quadratic formula to calculate the values of \( x \).
5. **Simplify the expression**: Perform the calculations to find \( x \).
### Example
Let's solve the quadratic equation \( 2x^2 - 4x - 6 = 0 \).
1. Identify \( a \), \( b \), and \( c \):
\[
a = 2, \quad b = -4, \quad c = -6
\]
2. Write the quadratic formula:
\[
x = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4 \cdot 2 \cdot (-6)}}}}{{2 \cdot 2}}
\]
3. Calculate the discriminant:
\[
(-4)^2 - 4 \cdot 2 \cdot (-6) = 16 + 48 = 64
\]
4. Substitute and simplify:
\[
x = \frac{{4 \pm \sqrt{64}}}{{4}} = \frac{{4 \pm 8}}{{4}}
\]
This gives two solutions:
\[
x_1 = \frac{{4 + 8}}{{4}} = \frac{12}{4} = 3
\]
\[
x_2 = \frac{{4 - 8}}{{4}} = \frac{-4}{4} = -1
\]
5. Therefore, the solutions to the equation \( 2x^2 - 4x - 6 = 0 \) are:
\[
x = 3 \quad \text{and} \quad x = -1
\]
And that's how you solve a quadratic equation using the quadratic formula!