Problem 177 - Elementary Algebra

Question

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=4 x^{2}-20 x+25 $$

Step-by-Step Solution

Answer

The x-intercept is (2.5, 0) and the y-intercept is (0, 25).

Step 1 - Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Substituting x = 0 into the equation gives: \( y = 4(0)^2 - 20(0) + 25 \)This simplifies to:\( y = 25 \)So, the y-intercept is (0, 25).

Step 2 - Find the x-intercepts

To find the x-intercepts, set y to 0 and solve for x. The equation becomes: \( 0 = 4x^2 - 20x + 25 \)This is a quadratic equation. Recall the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)Here, \( a = 4 \), \( b = -20 \), and \( c = 25 \).

Step 3 - Calculate the Discriminant

First, calculate the discriminant (\( b^2 - 4ac \)): \( (-20)^2 - 4(4)(25) \)Calculate \( 400 - 400 \ = 0 \)The discriminant is 0, which means there is exactly one real solution.

Step 4 - Use the Quadratic Formula

Since the discriminant is 0, substitute back into the quadratic formula: \( x = \frac{-(-20) \pm \sqrt{0}}{2(4)} \)Simplifying this gives: \( x = \frac{20}{8} \ = \frac{5}{2} \ = 2.5 \)So, the x-intercept is (2.5, 0).

Key Concepts

y-intercept

In a graph, the y-intercept is the point where the curve crosses the y-axis. This is the value of y when x equals 0. In our quadratic function example, we have the equation:
$$ y = 4x^2 - 20x + 25. $$
To find the y-intercept, we'll substitute x = 0 into the equation. This simplifies to:
$$ y = 4(0)^2 - 20(0) + 25. $$
Thus, the y-intercept is (0, 25). It means the graph passes through the point on the y-axis where y equals 25. This is an easy and quick way to find one of the key points of a quadratic graph.

x-intercept

The x-intercepts, also known as roots or zeros, are the points where the graph crosses the x-axis. To find the x-intercepts, we set y to 0 and solve for x. Given our equation:
$$ 0 = 4x^2 - 20x + 25. $$
This step transforms our equation into a quadratic equation to solve for x. Quadratic equations often yield two solutions, but they can sometimes have one or none. These intercepts give us essential insight into where the graph meets the x-axis, one of the crucial aspects when sketching or interpreting the quadratic function.

quadratic formula

The quadratic formula is a special formula used to solve quadratic equations of the form:
$$ ax^2 + bx + c = 0. $$
The formula is:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. $$
In our example, the values are:

• a = 4
• b = -20
• c = 25

Substituting these values into the formula helps us find the x-intercepts. This formula works regardless of the complexity of the quadratic equation and is a powerful tool in algebra.

discriminant

The discriminant is a part of the quadratic formula under the square root, given as:
$$ b^2 - 4ac. $$
It determines the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root.
• If negative, there are no real roots (the solutions would be complex numbers).

For our equation, the discriminant calculation is:
$$ (-20)^2 - 4(4)(25) = 400 - 400 = 0. $$
A discriminant of 0 means we have a double root or one unique real solution at (2.5, 0). Recognizing the value and meaning of the discriminant is crucial for understanding how many x-intercepts the graph has and where they lie on the x-axis.