Questions: Determine the x - and y-intercepts: y=-2 x^2-12 x-10. (If applicable, give your answer as a decimal.) x-intercepts: ) and y-intercept:

Determine the x - and y-intercepts: y=-2 x^2-12 x-10. (If applicable, give your answer as a decimal.) x-intercepts: ) and y-intercept:
Transcript text: Determine the $x$ - and $y$-intercepts: $y=-2 x^{2}-12 x-10$. (If applicable, give your answer as a decimal.) x-intercepts: $\square$ $\square$ ) and $\square$ $\square$ $y$-intercept: $\square$ $\square$
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Solution

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Solution Steps

To find the intercepts of the quadratic equation \( y = -2x^2 - 12x - 10 \), we need to determine where the graph of the equation crosses the x-axis and y-axis.

  1. X-intercepts: Set \( y = 0 \) and solve the quadratic equation \( -2x^2 - 12x - 10 = 0 \) for \( x \). This can be done using the quadratic formula.
  2. Y-intercept: Set \( x = 0 \) and solve for \( y \).
Step 1: Finding the X-Intercepts

To find the x-intercepts, we set \( y = 0 \) in the equation \( y = -2x^2 - 12x - 10 \). This leads to the equation: \[ -2x^2 - 12x - 10 = 0 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \), where \( D \) is the discriminant, we find: \[ D = b^2 - 4ac = (-12)^2 - 4(-2)(-10) = 144 - 80 = 64 \] Thus, the x-intercepts are: \[ x = \frac{12 \pm \sqrt{64}}{-4} = \frac{12 \pm 8}{-4} \] Calculating the two solutions gives: \[ x_1 = \frac{20}{-4} = -5.0 \quad \text{and} \quad x_2 = \frac{4}{-4} = -1.0 \]

Step 2: Finding the Y-Intercept

To find the y-intercept, we set \( x = 0 \) in the equation: \[ y = -2(0)^2 - 12(0) - 10 = -10 \]

Final Answer

The x-intercepts are \( x = -5.0 \) and \( x = -1.0 \), and the y-intercept is \( y = -10 \). Therefore, the final answers are: \[ \boxed{x = -5.0}, \quad \boxed{x = -1.0}, \quad \text{and} \quad \boxed{y = -10} \]

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