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Chapter 11: Problem 80

Find the \(x\) -intercepts of the function given by \(g(x)=3 x^{2}-5 x-1\).

Short Answer

Expert verified
The x-intercepts are \( x = \frac{5 + \sqrt{37}}{6} \) and \( x = \frac{5 - \sqrt{37}}{6} \).

Step by step solution

01

- Set the function equal to zero

To find the x-intercepts, set the function equal to zero: \[g(x) = 0\]Replace \( g(x) \) with the given function: \[3x^2 - 5x - 1 = 0\]
02

- Identify coefficients for the quadratic formula

The quadratic formula is used to find the roots of a quadratic equation \(ax^2 + bx + c = 0\): \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Since \(a = 3\), \(b = -5\), and \(c = -1\), plug these values into the quadratic formula.
03

- Calculate the discriminant

Compute the discriminant \(\Delta\) which is under the square root in the quadratic formula: \[\Delta = b^2 - 4ac\]Substitute the coefficients: \[\Delta = (-5)^2 - 4(3)(-1) = 25 + 12 = 37\]
04

- Apply the quadratic formula

Now that the discriminant is found, use the quadratic formula to find the roots:\[x = \frac{-(-5) \pm \sqrt{37}}{2(3)} = \frac{5 \pm \sqrt{37}}{6}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic equation
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.
quadratic formula
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.
discriminant
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:
\[x = \frac{-(-5) \pm \sqrt{37}}{2(3)} = \frac{5 \pm \sqrt{37}}{6}\]

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Most popular questions from this chapter

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (5,-6) $$

For each of the following, write the equation of the parabola that has the shape of \(f(x)=2 x^{2}\) or \(g(x)=-2 x^{2}\) and has a maximum value or \(a\) minimum value at the specified point. Minimum: \((2,0)\)

Review solving systems of three equations in three unknowns. Solve. $$ \begin{aligned} &z=-5\\\ &2 x-y+3 z=-27\\\ &x+2 y+7 z=-26 \end{aligned} $$

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (-4,-2) $$

Let \(f(x)=x^{2} .\) Find \(x\) such that \(f(x)=19\).
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