1) Solve `x^{2}-4x-5=0` and enter solutions below. please do correctly

Answer 1

To solve the quadratic equation \(x^{2}-4x-5=0\), we can use the quadratic formula:

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

where \(a = 1\), \(b = -4\), and \(c = -5\).

Now, first, we calculate the discriminant \(b^2 - 4ac\):

\[ b^2 - 4ac = (-4)^2 - 4(1)(-5) = 16 + 20 = 36 \]

Now we plug the values into the quadratic formula:

\[ x = \frac{-(-4) \pm \sqrt{36}}{2(1)} = \frac{4 \pm 6}{2} \]

This gives us two solutions:

1. \(x = \frac{4 + 6}{2} = \frac{10}{2} = 5\)
2. \(x = \frac{4 - 6}{2} = \frac{-2}{2} = -1\)

Thus, the solutions to the equation \(x^{2}-4x-5=0\) are:

\[ x = 5 \quad \text{and} \quad x = -1 \]

Answer 2

now graph it please

Answer 3

I can’t directly create graphical images. However, I can guide you through how to graph the quadratic equation \(y = x^2 - 4x - 5\).

Steps to Graph the Quadratic Function

1. Identify Key Features:

   • Vertex: The vertex formula for a quadratic \(y = ax^2 + bx + c\) is at \(x = -\frac{b}{2a}\).
     • Here, \(a = 1\) and \(b = -4\): \[ x = -\frac{-4}{2(1)} = \frac{4}{2} = 2 \]
   • To find the \(y\)-coordinate of the vertex, substitute \(x = 2\) into the equation: \[ y = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9 \]
   • So the vertex is at \((2, -9)\).

2. Find the Y-Intercept:

   • Set \(x = 0\): \[ y = 0^2 - 4(0) - 5 = -5 \]
   • The y-intercept is \((0, -5)\).

3. Find the X-Intercepts:

   • We already calculated the x-intercepts when solving the equation, which are \(x = 5\) and \(x = -1\). So the x-intercepts are \((5, 0)\) and \((-1, 0)\).

4. Plot Points:

   • Plot the vertex \((2, -9)\), y-intercept \((0, -5)\), and x-intercepts \((-1, 0)\) and \((5, 0)\).

5. Sketch the Parabola:

   • Draw the parabolic curve through the points plotted. Since the leading coefficient \(a = 1\) is positive, the parabola opens upwards.

Summary of Important Points:

• Vertex: \((2, -9)\)
• Y-Intercept: \((0, -5)\)
• X-Intercepts: \((-1, 0)\) and \((5, 0)\)

You can use graphing software or graphing calculators to draw a more accurate representation by inputting the equation \(y = x^2 - 4x - 5\).