x 2 3 13 in Math Algebra
The keyword x 2 3 13 is commonly shorthand for the quadratic expression \(x^2+3x+13\). A standard algebra question built from this keyword is to solve the quadratic equation \(x^2+3x+13=0\) and determine whether it has real solutions.
Interpretation used: “x 2 3 13” is treated as \(x^2+3x+13 = 0\), where \(a=1\), \(b=3\), and \(c=13\).
1) Identify the Discriminant
The discriminant is \(D=b^2-4ac\). For this equation: \[ D = 3^2 - 4(1)(13) = 9 - 52 = -43 \] Since \(D < 0\), there are no real solutions; the roots are a pair of complex conjugates.
2) Solve Using the Quadratic Formula
Applying the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\): \[ x = \frac{-3 \pm \sqrt{-43}}{2(1)} = \frac{-3 \pm i\sqrt{43}}{2} \]
| Feature | Value | Implication |
|---|---|---|
| Vertex | \((-1.5, 10.75)\) | Minimum point is far above the x-axis. |
| Discriminant | \(-43\) | The parabola never crosses the x-axis. |
| y-intercept | \((0, 13)\) | The value of the expression when \(x=0\). |
3) Visualization: Parabola for \(y=x^2+3x+13\)
Conclusion: The quadratic equation \(x^2+3x+13=0\) has no real solutions. The complex roots are \(x = \frac{-3 \pm i\sqrt{43}}{2}\).