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Chapter 2: Problem 1

Find the \(x\) -intercepts of the given function. $$ y=x^{2}-3 x+1 $$

Short Answer

Expert verified
The x-intercepts are \( x = \frac{3 + \sqrt{5}}{2} \) and \( \frac{3 - \sqrt{5}}{2} \).

Step by step solution

01

Set the function equal to zero

To find the x-intercepts, set the function equal to zero: \( y = x^2 - 3x + 1 \) becomes \( 0 = x^2 - 3x + 1 \).
02

Apply the quadratic formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where in this equation \(a = 1\), \(b = -3\), and \(c = 1\).
03

Calculate the discriminant

Calculate the discriminant \( b^2 - 4ac \): \( (-3)^2 - 4(1)(1) = 9 - 4 = 5 \).
04

Solve for x using the quadratic formula

Plug the value of the discriminant back into the quadratic formula to find the values of \(x\): \( x = \frac{-(-3) \pm \sqrt{5}}{2(1)} \) \( x = \frac{3 \pm \sqrt{5}}{2} \).
05

Simplify the solutions

The solutions can be simplified to \( x = \frac{3 + \sqrt{5}}{2} \) and \( x = \frac{3 - \sqrt{5}}{2} \).

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Key Concepts

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

x-intercepts
To find the x-intercepts of a function, you need to determine where the graph of the function crosses the x-axis. This happens when the value of y is zero. In other words, you set the function equal to zero and solve for x. By setting the function equal to zero, you can find the points where the curve meets the x-axis, giving you the x-intercepts.
quadratic formula
The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula allows you to find the solutions for x simply by plugging in the values of the coefficients a, b, and c from your quadratic equation. It is especially useful when the equation does not factorize easily.
discriminant
The discriminant is a component of the quadratic formula given by the expression \(b^2 - 4ac\). It provides critical information about the nature of the roots of a quadratic equation.
  • If the discriminant is positive, there are two distinct real solutions.
  • If it is zero, there is exactly one real solution.
  • If it is negative, there are no real solutions, but two complex solutions.
Calculating the discriminant helps predict the type of solutions you can expect before actually solving the equation.
solving quadratic equations
Solving quadratic equations involves finding the values of x that satisfy the equation \(ax^2 + bx + c = 0\). Here鈥檚 how to solve it step-by-step:
  • Set the equation to zero to form \(ax^2 + bx + c = 0\).
  • Identify the coefficients a, b, and c.
  • Calculate the discriminant \(b^2 - 4ac\).
  • Substitute a, b, and the discriminant into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • Simplify the equation to find the x-intercepts.
This method works universally for any quadratic equation, making it an essential technique in algebra.

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Most popular questions from this chapter

A savings and loan association estimates that the amount of money on deposit will be 1 million times the percentage rate of interest. For instance, a \(4 \%\) interest rate will generate $$\$ 4$$ million in deposits. If the savings and loan association can loan all the money it takes in at \(10 \%\) interest, what interest rate on deposits generates the greatest profit?

Consider a parabolic arch whose shape may be represented by the graph of \(y=9-x^{2}\), where the base of the arch lies on the \(x\) -axis from \(x=-3\) to \(x=3 .\) Find the dimensions of the rectangular window of maximum area that can be constructed inside the arch.

Find the dimensions of the rectangular garden of greatest area that can be fenced off (all four sides) with 300 meters of fencing.

Find the quadratic function \(f(x)=a x^{2}+b x+c\) that goes through \((2,0)\) and has a local maximum at \((0,1)\).

Find the \(x\) -intercepts of the given function. $$ y=2 x^{2}+5 x+2 $$
See all solutions

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