To determine the zero(s) of the quadratic function \( y = 2x^2 + 10x + 8 \), we can follow a systematic approach to find the values of \( x \) where \( y = 0 \). Here are the steps:
Step 1: Set the equation to zero
We first set the quadratic equation equal to zero: \[ 2x^2 + 10x + 8 = 0 \]
Step 2: Use the Quadratic Formula
We can solve for \( x \) using the Quadratic Formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] where \( a = 2 \), \( b = 10 \), and \( c = 8 \).
Step 3: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \): \[ b^2 = 10^2 = 100 \] \[ 4ac = 4 \cdot 2 \cdot 8 = 64 \] Now, calculate the discriminant: \[ b^2 - 4ac = 100 - 64 = 36 \]
Step 4: Substitute into the Quadratic Formula
Since the discriminant is positive, we have two real solutions: \[ x = \frac{{-10 \pm \sqrt{36}}}{2 \cdot 2} \]
Calculating the square root of the discriminant: \[ \sqrt{36} = 6 \] Now, substitute this back into the formula: \[ x = \frac{{-10 \pm 6}}{4} \]
Step 5: Find the two solutions
Now we calculate the two possible values for \( x \):
- For \( x = \frac{{-10 + 6}}{4} \): \[ x = \frac{{-4}}{4} = -1 \]
- For \( x = \frac{{-10 - 6}}{4} \): \[ x = \frac{{-16}}{4} = -4 \]
Step 6: List the zeros
The zeros of the quadratic function \( y = 2x^2 + 10x + 8 \) are: \[ x = -1 \quad \text{and} \quad x = -4 \]
Conclusion
Thus, the x-intercepts, or zeros of the function, where the graph crosses the x-axis, are \( x = -1 \) and \( x = -4 \).