Question

How do I graph the equation 2x^2 + 5x - 3 = 0?

Answer 1

To graph the quadratic equation \(2x^2 + 5x - 3 = 0\), you can follow these steps:

### 1. **Find the roots** using the quadratic formula:

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(2x^2 + 5x - 3 = 0\), we have:
- \(a = 2\)
- \(b = 5\)
- \(c = -3\)

First, calculate the discriminant \(D\):

\[
D = b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49
\]

Since \(D > 0\), there are two distinct real roots.

Now, substitute into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} = \frac{-5 \pm 7}{4}
\]

Calculating the two roots:

1. \(x_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}\)
2. \(x_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3\)

So the roots are \(x = \frac{1}{2}\) and \(x = -3\).

### 2. **Find the vertex**:

The vertex \(x\)-coordinate of a quadratic function in the form \(y = ax^2 + bx + c\) is given by:

\[
x = -\frac{b}{2a} = -\frac{5}{2 \cdot 2} = -\frac{5}{4}
\]

Now, substitute \(x\) into the original equation to find the \(y\)-coordinate of the vertex:

\[
y = 2\left(-\frac{5}{4}\right)^2 + 5\left(-\frac{5}{4}\right) - 3
\]
\[
= 2\left(\frac{25}{16}\right) - \frac{25}{4} - 3
\]
\[
= \frac{50}{16} - \frac{100}{16} - \frac{48}{16}
\]
\[
= \frac{50 - 100 - 48}{16} = \frac{-98}{16} = -\frac{49}{8}
\]

Thus, the vertex is \(\left(-\frac{5}{4}, -\frac{49}{8}\right)\).

### 3. **Plot points**:

You have the roots \(x = \frac{1}{2}\) and \(x = -3\), and the vertex \(\left(-\frac{5}{4}, -\frac{49}{8}\right)\). You might also want to plot a few other points on the graph for accuracy, by choosing values of \(x\) around the vertex and calculating corresponding \(y\) values.

### 4. **Draw the graph**:

- Mark the roots on the x-axis.
- Mark the vertex on the graph.
- The parabola opens upwards because the coefficient of \(x^2\) is positive.
- Draw a smooth curve through the points.

### Summary
In summary, to graph the quadratic function \(y = 2x^2 + 5x - 3\):

1. Mark the roots: \(x = \frac{1}{2}\) and \(x = -3\).
2. Mark the vertex: \(\left(-\frac{5}{4}, -\frac{49}{8}\right)\).
3. Plot additional points as needed.
4. Connect them with a smooth curve for the parabola.