Question
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Find the roots and the vertex of the quadratic on a calculator. Round all values to 3 decimal places (if necessary). \[ y=-x^{2}-10 x+35 \] Answer Attempti out of 2 Roots: \( \square \) and \( \square \) Vertex: \( \square \)

Ask by Chambers Nunez. in the United States
Feb 03,2025

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Tutor-Verified Answer

Answer

Roots: -12.746 and 2.746 Vertex: (-5.000, 60.000)

Solution

To find the **roots** and the **vertex** of the quadratic function: \[ y = -x^{2} - 10x + 35 \] we'll follow these steps: ### 1. Finding the Roots The roots of the quadratic equation \( y = 0 \) are the solutions to: \[ -x^{2} - 10x + 35 = 0 \] **Step-by-Step Calculation:** 1. **Rewrite the equation:** \[ -x^{2} - 10x + 35 = 0 \] 2. **Multiply by -1 to make it easier to apply the quadratic formula:** \[ x^{2} + 10x - 35 = 0 \] 3. **Apply the quadratic formula:** \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = 10 \), and \( c = -35 \). 4. **Calculate the discriminant:** \[ \Delta = b^2 - 4ac = 10^2 - 4(1)(-35) = 100 + 140 = 240 \] 5. **Find the square root of the discriminant:** \[ \sqrt{240} \approx 15.4919 \] 6. **Compute the roots:** \[ x = \frac{-10 \pm 15.4919}{2} \] - First root: \[ x = \frac{-10 + 15.4919}{2} \approx \frac{5.4919}{2} \approx 2.746 \] - Second root: \[ x = \frac{-10 - 15.4919}{2} \approx \frac{-25.4919}{2} \approx -12.746 \] ### 2. Finding the Vertex The vertex of a quadratic function \( y = ax^2 + bx + c \) is given by: \[ \left( -\frac{b}{2a}, \, y\left(-\frac{b}{2a}\right) \right) \] **Step-by-Step Calculation:** 1. **Determine the x-coordinate of the vertex:** \[ x = -\frac{b}{2a} = -\frac{-10}{2(-1)} = \frac{10}{-2} = -5 \] 2. **Find the y-coordinate by substituting \( x = -5 \) back into the original equation:** \[ y = -(-5)^2 - 10(-5) + 35 \] \[ y = -25 + 50 + 35 \] \[ y = 60 \] So, the vertex is at \( (-5, 60) \). ### **Final Answer:** - **Roots:** \( -12.746 \) and \( 2.746 \) - **Vertex:** \( (-5.000,\ 60.000) \)

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Beyond the Answer

To find the roots of the quadratic \( y = -x^2 - 10x + 35 \), you can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = -1 \), \( b = -10 \), and \( c = 35 \). Plugging these values into the formula gives you the roots approximately as \( x = 2.410 \) and \( x = -14.410 \). For the vertex of the quadratic, use the formula \( x = -\frac{b}{2a} \). Substituting the coefficients, you will find that the x-coordinate of the vertex is \( x = -5 \). To find the corresponding y-coordinate, substitute \( x = -5 \) back into the original equation, yielding \( y = 60 \). Therefore, the vertex is \( (-5, 60) \).

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