Question

solving quadratic equations using quadratic formula

Answer 1

The quadratic formula is used to solve quadratic equations of the 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 quadratic 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 \( D = b^2 - 4ac \). This value will help determine the nature of the roots.
- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is exactly one real root (or a repeated root).
- If \( D < 0 \), there are two complex roots.

3. **Apply the quadratic formula**: Substitute \( a \), \( b \), and \( D \) into the formula to find the values of \( x \):
- Use the positive square root for one solution: \( x_1 = \frac{-b + \sqrt{D}}{2a} \)
- Use the negative square root for the other solution: \( x_2 = \frac{-b - \sqrt{D}}{2a} \)

### Example

Let's solve the quadratic equation \( 2x^2 + 3x - 5 = 0 \):

1. **Identify coefficients**:
- \( a = 2 \)
- \( b = 3 \)
- \( c = -5 \)

2. **Calculate the discriminant**:
\[
D = b^2 - 4ac = 3^2 - 4 \cdot 2 \cdot (-5) = 9 + 40 = 49
\]

3. **Apply the quadratic formula**:
\[
x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{2 \cdot 2}
\]
\[
x = \frac{-3 \pm 7}{4}
\]

Now calculate the two possible values for \( x \):
- For \( x_1 \):
\[
x_1 = \frac{-3 + 7}{4} = \frac{4}{4} = 1
\]

- For \( x_2 \):
\[
x_2 = \frac{-3 - 7}{4} = \frac{-10}{4} = -2.5
\]

### Solutions
Thus, the solutions to the equation \( 2x^2 + 3x - 5 = 0 \) are:
\[ x_1 = 1 \]
\[ x_2 = -2.5 \]

You can use this method for any quadratic equation to find the roots!