Problem 9 Find the $x$ -intercepts for t... [FREE SOLUTION]

Chapter 9: Problem 9

Find the $x$ -intercepts for the parabola whose equation is given. If the $x$ -intercepts are irrational numbers, round your answers to the nearest tenth. $$y=x^{2}+2 x-4$$

Short Answer

Expert verified

The $x$-intercepts of the given parabola are $x = 1.2$ and $x = -3.2$.

Step by step solution

Convert given equation

Set $y$ equal to 0 to convert the equation into quadratic form. This will provide the equation $0=x^{2}+2x-4$.

Apply Quadratic Formula

Rearrange the equation in terms of $x$ using the quadratic formula, $x=-b\pm\sqrt{b^2-4ac} / 2a$, where $a=1$, $b=2$, and $c=-4$. Substitution of the given values into the formula gives us $x= \frac{-2\pm\sqrt{(2)^2-4(1)(-4)}}{2(1)}$.

Simplify the square root

Calculate the value under the square root (the discriminant) to get 20. The equation becomes $x= \frac{-2\pm\sqrt{20}}{2}$.

Solve for $x$

Simplify and solve for $x$ to find $x= -1\pm\sqrt{5}$. These are the exact values, but since we are asked for irrational $x$-intercepts to the nearest tenth, we need to further simplify and round our answers.

Approximate to Rational Numbers

Approximate to the nearest tenth to get the final solution: $x = -1 + \sqrt{5}$ is about 1.2 and $x = -1 - \sqrt{5}$ is about -3.2. Therefore, the $x$-intercepts are at $x = 1.2$ and $x = -3.2$.

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Key Concepts

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

Parabola

A parabola is a unique curve, usually represented on a graph by a quadratic equation, like the one in our exercise, given by $y = x^2 + 2x - 4$. This function is specifically a type of polynomial equation, which graphically forms a 'U' or an inverted 'U' shape. Understanding the structure of a parabola helps us know the nature of its intercepts and vertices.
A parabola has several distinctive parts that you should know about:

Vertex: The turning point of the parabola. In our case, it's influenced by the coefficients of our quadratic equation.
Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror images.
Direction: Parabolas can open upwards or downwards depending on the sign of the leading coefficient ($a$). For $y = x^2 + 2x - 4$, since $a$ is 1, an upward opening is indicated.
X-intercepts: These points, where the function crosses the x-axis, are our main focus here, requiring us to solve the equation $x^2 + 2x - 4 = 0$.

Understanding these features allows us to solve and graph quadratic functions effectively and predict how changes in the equation affect the graph.

Quadratic Formula

The Quadratic Formula is a powerful tool for finding the roots of any quadratic equation of the form $ax^2 + bx + c = 0$. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula derives solutions by calculating the values where the expression equals zero, which determines the x-intercepts of the parabola.
Using the quadratic formula involves several stages:

Identify $a, b,$ and $c$: These are coefficients from the equation. In our case, $a = 1$, $b = 2$, and $c = -4$.
Calculate the Discriminant: The expression under the square root, $b^2 - 4ac$, is vital as it reveals the nature of the roots. A positive discriminant suggests two real roots, a zero indicates one, while a negative predicts complex roots. Here, $(2)^2 - 4(1)(-4) = 20$ means two distinct real roots.
Apply the Formula: Place the values into the formula to solve for $x$. For instance, we calculated $x= \frac{-2 \pm \sqrt{20}}{2}$.
Simplify: Perform calculations accurately to reach an exact or approximate solution as required.

Mastering the quadratic formula equips you to efficiently tackle any quadratic without needing to factor manually all the time.

X-intercepts

X-intercepts are crucial points where the graph of a function touches or crosses the x-axis. In the equation $y = x^2 + 2x - 4$, finding these intercepts requires setting $y$ to zero, resulting in a quadratic form that can be solved with the quadratic formula.
When solving for x-intercepts:

Set the Equation to Zero: Transform the equation to $x^2 + 2x - 4 = 0$ to identify where the parabola intersects the x-axis.
Use the Quadratic Formula: Plug into the quadratic formula to find potential roots, which are the x-coordinates of the intercepts.
Exact and Approximate Solutions: For our example, solving yields exact values $x = -1 \pm \sqrt{5}$. Further approximation gives the rounded values $x = 1.2$ and $x = -3.2$, indicating intercepts on the graph.

X-intercepts are essential for graphing and understanding the function's behavior, as they offer insights into how the function behaves across real numbers. Recognizing and solving for these intercepts aides in visualizing and predicting the shape of the parabola effectively.

91影视

Find the \(x\) -intercepts for the parabola whose equation is given. If the \(x\) -intercepts are irrational numbers, round your answers to the nearest tenth. $$y=x^{2}+2 x-4$$

Short Answer

Step by step solution

Convert given equation

Apply Quadratic Formula

Simplify the square root

Solve for \(x\)

Approximate to Rational Numbers

Key Concepts

Parabola

Quadratic Formula

X-intercepts

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