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Chapter 11: Problem 77

Find the \(x\) -intercepts of the function given by \(f(x)=4 x^{2}+2 x-3\).

Short Answer

Expert verified
The x-intercepts are respectively found when simplified for both positive and negative values using quadratic substitutions.

Step by step solution

01

Identify the Quadratic Function

The given function is a quadratic function of the form .. It has the coefficients: Suggesting we are dealing with quadratic function
02

Set the Equation to Zero

To find the x-intercepts, set the function equal to zero: .
03

Apply the Quadratic Formula

Use the quadratic formula: used which the values and x represents the needed formula to find elementary quadratic equations.
04

Calculate the Discriminant

First, calculate the discriminant: which represents the delta values from the quadratic form to solve given factors
05

Solve for x using the Quadratic Formula

Substitute the values of a, b, and c into the quadratic formula: which divides as needed to find the respective substituted value
06

Simplify

Simplify the expressions under the square root, and solve for both positive and negative values of x: respectively found which would give our final needed value.
07

State the x-intercepts

Hence, the x-intercepts of the function are found using the correctly termed calculations.

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Key Concepts

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

quadratic formula
A quadratic equation is an equation of the form \[ ax^2 + bx + c = 0 \]
where
  • \(a\) is the coefficient of the squared term,
  • \(b\) is the coefficient of the linear term,
  • \(c\) is the constant term.
The quadratic formula is a tool to solve such an equation. It states that the solutions for \(x\) can be found using: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows us to find the roots, or x-intercepts, of quadratic equations without the need to factorize them directly.
discriminant
The discriminant is a component of the quadratic formula which appears under the square root symbol, \[ b^2 - 4ac \]. The value of the discriminant helps us determine the nature of the roots for the quadratic equation.
  • If the discriminant is positive \((b^2 - 4ac > 0)\), the equation has two distinct real roots.
  • If the discriminant is zero \((b^2 - 4ac = 0)\), the equation has exactly one real root (also called a repeated root).
  • If the discriminant is negative \((b^2 - 4ac < 0)\), the equation has no real roots, but two complex roots.
This insight is useful as it tells us how many solutions to expect when solving the quadratic equation.
x-intercepts
The x-intercepts of a function are the points where the graph of the function crosses the x-axis. To find the x-intercepts of a quadratic function, we need to find the values of \(x\) for which the function equals zero. This means solving the equation \[ f(x) = ax^2 + bx + c = 0 \]. Once we find the roots (solutions) of this equation using the quadratic formula or other methods, we get the x-intercepts. Supposing we have found them as \(x_1\) and \(x_2\), the x-intercepts will be the points \((x_1, 0)\) and \((x_2, 0)\).
solving quadratic equations
To solve quadratic equations, one can use various methods aside from the quadratic formula. Some other common methods include:
  • **Factoring**: Express the quadratic equation as the product of two binomials. For example, \[ x^2 - 5x + 6 = (x-2)(x-3) = 0 \].
    Then, solve for \(x\) by setting each binomial to zero.
  • **Completing the Square**: Rewriting the equation to make one side a perfect square trinomial. This involves adding and then subtracting the same value to keep the equation balanced.
  • **Graphing**: Plotting the quadratic function and identifying the points where it intersects the x-axis.
Using these methods helps to provide a deeper understanding of the quadratic equation's roots and how they relate to the function's graph. However, the quadratic formula is often the most straightforward and reliable method for finding the x-intercepts, especially for complex or non-factorable quadratic equations.

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Most popular questions from this chapter

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

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. $$ (-2,-5) $$

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}-10 x-22 $$

Solve. $$ 9 x^{2}-16=0 $$

Replace the blanks in each equation with constants to complete the square and form a true equation. \(x^{2}+8 x+\)_____\(=(x+\)_____\()^{2}\)
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