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Chapter 11: Problem 57

Find any \(x\) -intercepts and the \(y\) -intercept. If no \(x\) -intercepts exist, state this. $$ g(x)=x^{2}+x-5 $$

Short Answer

Expert verified
The y-intercept is (0, -5). The x-intercepts are \( \left( \frac{-1+\sqrt{21}}{2}, 0 \right) \) and \( \left( \frac{-1-\sqrt{21}}{2}, 0 \right) \).

Step by step solution

01

Title - Find the y-intercept

The y-intercept occurs where the graph crosses the y-axis, which is at x=0. To find it, substitute x=0 into the function g(x):\[ g(0) = 0^2 + 0 - 5 = -5 \]. Therefore, the y-intercept is \( (0, -5) \).
02

Title - Find the x-intercepts

The x-intercepts occur where the graph crosses the x-axis, which is at y=0 (i.e., where g(x)=0). Set the function equal to zero and solve for x: \[ x^2 + x - 5 = 0 \].
03

Title - Use the quadratic formula

To solve the quadratic equation, use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. For our equation \( x^2 + x - 5 = 0 \), the coefficients are: a=1, b=1, and c=-5. Substitute these values: \[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 + 20}}{2 \cdot 1} = \frac{-1 \pm \sqrt{21}}{2} \].
04

Title - Simplify the solutions

Simplify the expression to find the exact values of the x-intercepts: \[ x = \frac{-1+\sqrt{21}}{2} \] and \[ x = \frac{-1-\sqrt{21}}{2} \]. Therefore, the x-intercepts are \( \left( \frac{-1+\sqrt{21}}{2}, 0 \right) \) and \( \left( \frac{-1-\sqrt{21}}{2}, 0 \right) \).

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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, we look for the points where the graph intersects the x-axis. This occurs where the value of y is zero. If we have a function like g(x) = x^2 + x - 5, we set it equal to zero and solve for x: \[ x^2 + x - 5 = 0 \] By solving this equation, we can find the x-values (x-intercepts). These are the points where the function equals zero.
y-intercept
The y-intercept is the point where the graph crosses the y-axis. This happens when the value of x is zero. To find the y-intercept of a function, substitute x = 0 into the equation of the function. For example, for g(x) = x^2 + x - 5 when x=0: \[ g(0) = 0^2 + 0 - 5 = -5 \] Hence, the y-intercept is (0, -5).
quadratic formula
The quadratic formula is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0. The formula is: \[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \] For the equation x^2 + x - 5 = 0, where a=1, b=1, and c=-5, we can substitute these values into the formula: \[ x = \frac{-1 \, \pm \, \sqrt{1^2 - 4 \, \cdot \, 1 \, \cdot \, (-5)}}{2 \, \cdot \, 1} = \frac{-1 \, \pm \, \sqrt{1 + 20}}{2} = \frac{-1 \, \pm \, \sqrt{21}}{2} \] This gives the solutions for x.
solving quadratic equations
Solving quadratic equations typically involves finding the values of x that make the equation true. Common methods include factoring, completing the square, and using the quadratic formula.
  • Factoring: We try to express the equation in the form (x - p)(x - q) = 0.
  • Completing the Square: We rewrite the equation in the form (x - h)^2 = k.
  • Quadratic Formula: We use the formula \[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \] when other methods are complicated or infeasible.
In our example g(x) = x^2 + x - 5, we used the quadratic formula to find the x-values where the function equals zero, providing the x-intercepts.

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Most popular questions from this chapter

Replace the blanks in each equation with constants to complete the square and form a true equation. \(x^{2}+\frac{2}{5} x+\)_____\(=(x+\)_____\()^{2}\)

To prepare for Section \(11.2,\) review evaluating expressions and simplifying radical expressions (Sections \(1.8,10.3\) and \(10.8)\) Evaluate. [ 1.8] $$ b^{2}-4 a c, \text { for } a=1, b=-1, \text { and } c=4 $$

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ g(x)=x^{2}+5 x+2 $$

For each equation under the given condition, (a) find \(k\) and (b) find the other solution. \(x^{2}-(6+3 i) x+k=0 ;\) one solution is 3

Solve by completing the square. Show your work. $$ x^{2}+8 x=9 $$
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