Finding The Zeros Of A Quadratic Function

Finding the zeros of a quadratic function is a fundamental skill in algebra, unlocking a deeper understanding of these ubiquitous mathematical expressions. The zeros, also known as roots or x-intercepts, are the values of 'x' that make the quadratic function equal to zero. These points represent where the parabola, the graphical representation of a quadratic function, intersects the x-axis. Mastering the techniques to find these zeros is essential for solving various problems in mathematics, physics, engineering, and other fields.

Understanding Quadratic Functions

Before diving into the methods for finding zeros, it's crucial to understand what a quadratic function is. A quadratic function is a polynomial function of degree two, generally expressed in the standard form:

f(x) = ax² + bx + c

where 'a', 'b', and 'c' are constants, and 'a' ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if 'a' is positive and downwards if 'a' is negative.

'a': Determines the direction and "width" of the parabola.
'b': Affects the position of the parabola's vertex.
'c': Represents the y-intercept of the parabola (the point where the parabola intersects the y-axis).

The zeros of a quadratic function are the solutions to the equation:

ax² + bx + c = 0

These zeros can be real or complex numbers, and a quadratic function can have two distinct real zeros, one repeated real zero, or two complex zeros.

Methods for Finding the Zeros

There are several methods for finding the zeros of a quadratic function, each with its strengths and weaknesses:

Factoring: This method involves expressing the quadratic expression as a product of two linear factors.
Completing the Square: This technique transforms the quadratic equation into a perfect square trinomial, making it easier to solve.
Quadratic Formula: This is a general formula that provides the zeros directly from the coefficients 'a', 'b', and 'c'.

Let's explore each method in detail.

1. Factoring

Factoring is the most straightforward method when it works. It involves rewriting the quadratic expression as a product of two linear factors. If we can find two numbers that satisfy certain conditions, we can easily determine the zeros.

Steps for Factoring:

Step 1: Write the quadratic equation in standard form: Ensure the equation is in the form ax² + bx + c = 0.
Step 2: Find two numbers that multiply to 'ac' and add up to 'b'. Let's call these numbers 'p' and 'q'. So, p * q = ac and p + q = b.
Step 3: Rewrite the middle term using these numbers: Replace 'bx' with 'px + qx'. The equation now becomes ax² + px + qx + c = 0.
Step 4: Factor by grouping: Factor out the greatest common factor (GCF) from the first two terms and the last two terms separately.
Step 5: Write the expression as a product of two factors: You should now have an expression of the form (Ax + B)(Cx + D) = 0.
Step 6: Set each factor equal to zero and solve for 'x': Solve Ax + B = 0 and Cx + D = 0 to find the zeros.

Example 1:

Find the zeros of the quadratic function f(x) = x² + 5x + 6.

Step 1: The equation is already in standard form.
Step 2: We need two numbers that multiply to 6 (1 * 6) and add up to 5. These numbers are 2 and 3.
Step 3: Rewrite the middle term: x² + 2x + 3x + 6 = 0.
Step 4: Factor by grouping: x(x + 2) + 3(x + 2) = 0.
Step 5: Write as a product of two factors: (x + 2)(x + 3) = 0.
Step 6: Set each factor to zero:
- x + 2 = 0 => x = -2
- x + 3 = 0 => x = -3

Therefore, the zeros of the quadratic function are x = -2 and x = -3.

Example 2:

Find the zeros of the quadratic function f(x) = 2x² - 5x - 3.

Step 1: The equation is already in standard form.
Step 2: We need two numbers that multiply to -6 (2 * -3) and add up to -5. These numbers are -6 and 1.
Step 3: Rewrite the middle term: 2x² - 6x + x - 3 = 0.
Step 4: Factor by grouping: 2x(x - 3) + 1(x - 3) = 0.
Step 5: Write as a product of two factors: (2x + 1)(x - 3) = 0.
Step 6: Set each factor to zero:
- 2x + 1 = 0 => x = -1/2
- x - 3 = 0 => x = 3

Therefore, the zeros of the quadratic function are x = -1/2 and x = 3.

Limitations of Factoring:

Factoring is not always possible, especially when the zeros are irrational or complex numbers. In such cases, other methods like completing the square or the quadratic formula are more appropriate.

2. Completing the Square

Completing the square is a technique that transforms a quadratic expression into a perfect square trinomial. This method is useful when factoring is difficult or impossible, and it also provides a way to derive the quadratic formula.

Steps for Completing the Square:

Step 1: Write the quadratic equation in standard form: Ensure the equation is in the form ax² + bx + c = 0.
Step 2: Divide the equation by 'a' (if a ≠ 1): This step makes the coefficient of x² equal to 1. The equation becomes x² + (b/a)x + (c/a) = 0.
Step 3: Move the constant term to the right side of the equation: Subtract (c/a) from both sides: x² + (b/a)x = -(c/a).
Step 4: Add the square of half the coefficient of 'x' to both sides: Take half of the coefficient of 'x', which is (b/2a), and square it: (b/2a)². Add this to both sides of the equation: x² + (b/a)x + (b/2a)² = -(c/a) + (b/2a)².
Step 5: Rewrite the left side as a perfect square: The left side is now a perfect square trinomial: (x + b/2a)² = -(c/a) + (b/2a)².
Step 6: Simplify the right side: Find a common denominator and simplify the expression on the right side.
Step 7: Take the square root of both sides: Remember to include both positive and negative square roots.
Step 8: Solve for 'x': Isolate 'x' to find the zeros.

Example:

Find the zeros of the quadratic function f(x) = x² + 6x + 5.

Step 1: The equation is already in standard form.
Step 2: The coefficient of x² is already 1.
Step 3: Move the constant term to the right side: x² + 6x = -5.
Step 4: Add the square of half the coefficient of 'x' to both sides: Half of 6 is 3, and 3² is 9. So, x² + 6x + 9 = -5 + 9.
Step 5: Rewrite the left side as a perfect square: (x + 3)² = 4.
Step 6: Take the square root of both sides: x + 3 = ±2.
Step 7: Solve for 'x':
- x + 3 = 2 => x = -1
- x + 3 = -2 => x = -5

Therefore, the zeros of the quadratic function are x = -1 and x = -5.

Advantages of Completing the Square:

Works for all quadratic equations, even those that are difficult to factor.
Provides a method for deriving the quadratic formula.
Helps to understand the vertex form of a quadratic equation.

Disadvantages of Completing the Square:

Can be more time-consuming than factoring or using the quadratic formula.
Involves working with fractions, which can be cumbersome.

3. Quadratic Formula

The quadratic formula is a general formula that provides the zeros of a quadratic function directly from its coefficients. It is derived by completing the square on the general quadratic equation ax² + bx + c = 0.

The Quadratic Formula:

For a quadratic equation in the form ax² + bx + c = 0, the zeros are given by:

x = (-b ± √(b² - 4ac)) / 2a

Steps for Using the Quadratic Formula:

Step 1: Write the quadratic equation in standard form: Ensure the equation is in the form ax² + bx + c = 0.
Step 2: Identify the coefficients 'a', 'b', and 'c'.
Step 3: Substitute the values of 'a', 'b', and 'c' into the quadratic formula.
Step 4: Simplify the expression.
Step 5: Calculate the two possible values of 'x' using the plus and minus signs.

Example:

Find the zeros of the quadratic function f(x) = 2x² + 3x - 5.

Step 1: The equation is already in standard form.
Step 2: Identify the coefficients: a = 2, b = 3, c = -5.
Step 3: Substitute the values into the quadratic formula:

x = (-3 ± √(3² - 4 * 2 * -5)) / (2 * 2)
Step 4: Simplify the expression:

x = (-3 ± √(9 + 40)) / 4 x = (-3 ± √49) / 4 x = (-3 ± 7) / 4
Step 5: Calculate the two possible values of 'x':
- x = (-3 + 7) / 4 = 4 / 4 = 1
- x = (-3 - 7) / 4 = -10 / 4 = -5/2

Therefore, the zeros of the quadratic function are x = 1 and x = -5/2.

The Discriminant:

The expression inside the square root in the quadratic formula, b² - 4ac, is called the discriminant. The discriminant provides information about the nature of the zeros:

If b² - 4ac > 0: The quadratic function has two distinct real zeros.
If b² - 4ac = 0: The quadratic function has one repeated real zero.
If b² - 4ac < 0: The quadratic function has two complex zeros (conjugate pairs).

Advantages of the Quadratic Formula:

Works for all quadratic equations, regardless of whether they can be factored or not.
Provides a direct method for finding the zeros.
The discriminant provides information about the nature of the zeros.

Disadvantages of the Quadratic Formula:

Can be more complex to use than factoring, especially when the coefficients are simple.
Requires careful attention to signs and order of operations.

Choosing the Right Method

The choice of which method to use depends on the specific quadratic equation and personal preference:

Factoring: Use this method when the quadratic expression is easily factorable. It is the quickest and simplest method when it works.
Completing the Square: Use this method when factoring is difficult or impossible, and you want to understand the structure of the quadratic equation.
Quadratic Formula: Use this method when factoring is difficult or impossible, and you need a direct and reliable way to find the zeros.

In many cases, the quadratic formula is the most versatile and reliable method, as it works for all quadratic equations. However, mastering factoring and completing the square provides a deeper understanding of quadratic functions and their properties.

Examples and Applications

Let's look at some more examples and applications of finding the zeros of quadratic functions.

Example 1: Projectile Motion

A ball is thrown vertically upwards with an initial velocity of 20 m/s from a height of 2 meters. The height of the ball above the ground at time 't' seconds is given by the equation:

h(t) = -5t² + 20t + 2

Find the time when the ball hits the ground.

To find when the ball hits the ground, we need to find the time 't' when h(t) = 0. So we need to solve the quadratic equation:

-5t² + 20t + 2 = 0

We can use the quadratic formula to find the zeros:

t = (-20 ± √(20² - 4 * -5 * 2)) / (2 * -5) t = (-20 ± √(400 + 40)) / -10 t = (-20 ± √440) / -10 t = (-20 ± 2√110) / -10 t = (10 ± √110) / 5

We have two possible values for 't':

t = (10 + √110) / 5 ≈ 4.098 seconds
t = (10 - √110) / 5 ≈ -0.098 seconds

Since time cannot be negative, the ball hits the ground at approximately t = 4.098 seconds.

Example 2: Revenue Maximization

A company sells a product for 'p' dollars each. The number of units sold 'x' is related to the price by the equation:

x = 1000 - 20p

The revenue 'R' is given by the product of the price and the number of units sold:

R = px = p(1000 - 20p) = 1000p - 20p²

Find the price that maximizes the revenue.

To find the price that maximizes revenue, we need to find the vertex of the quadratic function R(p) = -20p² + 1000p. The x-coordinate (in this case, the p-coordinate) of the vertex is given by:

p = -b / 2a = -1000 / (2 * -20) = -1000 / -40 = 25

So, the price that maximizes revenue is p = $25.

Example 3: Bridge Design

The shape of a suspension bridge cable can be modeled by a quadratic function. Suppose the height of the cable above the road is given by:

y = 0.01x² - 0.5x + 10

where 'x' is the horizontal distance from one end of the bridge. Find the points where the cable touches the road (i.e., find the zeros of the function).

To find where the cable touches the road, we need to find the values of 'x' when y = 0. So we need to solve the quadratic equation:

0.01x² - 0.5x + 10 = 0

We can use the quadratic formula to find the zeros:

x = (0.5 ± √((-0.5)² - 4 * 0.01 * 10)) / (2 * 0.01) x = (0.5 ± √(0.25 - 0.4)) / 0.02 x = (0.5 ± √(-0.15)) / 0.02

Since the discriminant is negative, the quadratic function has two complex zeros. This means that the cable does not actually touch the road in this model. The cable is always above the road.

Conclusion

Finding the zeros of a quadratic function is a vital skill in mathematics with far-reaching applications. Whether you choose to factor, complete the square, or use the quadratic formula, understanding these methods empowers you to solve a wide range of problems. Remember to consider the strengths and weaknesses of each method and choose the one that best suits the specific problem at hand. By mastering these techniques, you will unlock a deeper understanding of quadratic functions and their role in the world around us.