Problem 80 Find the \(x\) -intercepts of th... [FREE SOLUTION] (2024)

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A quadratic equation is a type of polynomial equation of the form:
\[ax^2 + bx + c = 0\]
where:

• \(a\), \(b\), and \(c\) are coefficients
• x is the variable

The highest exponent of the variable is 2, which makes it a second-degree polynomial.
In our exercise, the quadratic equation given is \(3x^2 - 5x - 1 = 0\).
We set the equation equal to zero to find the x-intercepts, where the graph crosses the x-axis. This means we are looking for values of \(x\) that make the equation true.

The quadratic formula is a tool used to solve quadratic equations. It provides the solutions (roots) for the equation:
\[ax^2 + bx + c = 0\]
The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

To use the quadratic formula, follow these steps:

1. Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation
2. Calculate the discriminant \(\Delta = b^2 - 4ac\)
3. Plug the values of the coefficients and the discriminant into the formula
4. Simplify to find the values of \(x\)

In our example:

• \(a = 3\)
• \(b = -5\)
• \(c = -1\)

We substitute these values into the quadratic formula to find \(x\). This will give us the x-intercepts of the function.

The discriminant \(\Delta\) is a part of the quadratic formula, located under the square root:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
It is calculated as:
\[\Delta = b^2 - 4ac\]

The discriminant tells us about the nature of the roots:

• If \(\Delta > 0\), there are two distinct real roots
• If \(\Delta = 0\), there is exactly one real root
• If \(\Delta < 0\), there are no real roots, but two complex roots

In our exercise:
\[\Delta = (-5)^2 - 4(3)(-1) = 25 + 12 = 37\]
Since \(\Delta = 37\) and it is greater than zero, we know there are two distinct real roots.
These roots represent the x-intercepts of the function. Plugging \(\Delta\) back into the quadratic formula gives us:
\[x = \frac{-(-5) \pm \sqrt{37}}{2(3)} = \frac{5 \pm \sqrt{37}}{6}\]