How to Find the Zeros of a Quadratic Function: A Student Guide
Quick Answer
To find the zeros of a quadratic function like y = -3x² + 12x + 15, set y equal to zero and solve for x. The zeros are the x-values where the graph crosses the x-axis, which are (5, 0) and (-1, 0) in this case.
Finding the zeros of a quadratic function is an essential skill in algebra. The zeros, also known as roots or x-intercepts, are the points where the graph of the function intersects the x-axis. This means at these points, the value of y is zero. In the given equation, y = -3x² + 12x + 15, we need to determine where y equals zero.
### Steps to Find Zeros
1. **Set y to 0**: Start by setting the equation to zero:
0 = -3x² + 12x + 15
2. **Rearrange the Equation**: Move all terms to one side of the equation (this is already done in this case).
3. **Simplify the Equation**: To simplify calculations, divide the entire equation by -3:
0 = x² - 4x - 5
4. **Factor the Quadratic**: Now, we need to factor the quadratic expression. Look for two numbers that multiply to -5 and add up to -4. These numbers are -5 and +1, so we can write:
x² - 4x - 5 = (x - 5)(x + 1)
5. **Solve for x**: Set each factor equal to zero:
x - 5 = 0 → x = 5
x + 1 = 0 → x = -1
Thus, the zeros of the function are at x = 5 and x = -1. We can express these as points on the coordinate plane: (5, 0) and (-1, 0).
### Understanding Zeros vs. Y-Intercepts
It's important to note that zeros refer specifically to the x-axis crossings of the graph. In contrast, the y-intercept is the point where the graph crosses the y-axis, which occurs when x = 0. In this case, to find the y-intercept, we substitute x = 0 into the original equation:
y = -3(0)² + 12(0) + 15 = 15
So the y-intercept is (0, 15). Remember, when asked about zeros, you only count the points where y = 0.
### Real-World Applications
Understanding how to find zeros is crucial not just in pure mathematics but also in various real-world applications. For example, in physics, the zeros of a motion equation can represent the points in time when an object is at rest. In economics, zeros in profit functions can indicate break-even points.
By mastering the concept of zeros, you will find it easier to analyze and interpret the behavior of quadratic functions in different contexts. Keep practicing with different equations to strengthen your skills!
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