Determine The \[$y\$\]-intercept Of The Following Equation:\[$(x-5)(x+2)=y\$\]Answer:A. \[$(0,10)\$\], \[$(5,0)\$\], And \[$(-2,0)\$\]B. \[$(0,-10)\$\], \[$(-5,0)\$\], And \[$(-2,0)\$\]C.

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Introduction

In algebra, the y-intercept of an equation is the point at which the graph of the equation intersects the y-axis. It is a crucial concept in understanding the behavior of functions and their graphs. In this article, we will determine the y-intercept of the given equation (x-5)(x+2)=y.

Understanding the Equation

The given equation is a quadratic equation in the form of (x-a)(x-b)=y, where a and b are constants. To find the y-intercept, we need to find the values of x that make the equation true when y=0.

Step 1: Expand the Equation

To find the y-intercept, we first need to expand the given equation. We can do this by multiplying the two binomials:

(x-5)(x+2) = x^2 + 2x - 5x - 10 = x^2 - 3x - 10

Step 2: Set y=0

Now that we have expanded the equation, we can set y=0 and solve for x:

x^2 - 3x - 10 = 0

Step 3: Factor the Quadratic Equation

We can factor the quadratic equation as:

(x-5)(x+2) = 0

Step 4: Solve for x

Now that we have factored the quadratic equation, we can solve for x by setting each factor equal to 0:

x-5 = 0 --> x = 5 x+2 = 0 --> x = -2

Step 5: Find the y-intercept

Now that we have found the values of x, we can find the corresponding values of y by plugging these values back into the original equation:

For x = 5, we have: (5-5)(5+2) = 0 = 0

For x = -2, we have: (-2-5)(-2+2) = 0 = 0

For x = 0, we have: (0-5)(0+2) = -10 = -10

Conclusion

In conclusion, the y-intercept of the given equation (x-5)(x+2)=y is (0,-10), (-5,0), and (-2,0).

Answer

The correct answer is B. (0,-10), (-5,0), and (-2,0).

Discussion

The y-intercept of an equation is an important concept in algebra. It is used to determine the behavior of functions and their graphs. In this article, we have determined the y-intercept of the given equation (x-5)(x+2)=y. We have expanded the equation, set y=0, factored the quadratic equation, solved for x, and found the corresponding values of y. The y-intercept is an essential concept in algebra, and it is used in a variety of applications, including physics, engineering, and economics.

Key Takeaways

  • The y-intercept of an equation is the point at which the graph of the equation intersects the y-axis.
  • To find the y-intercept, we need to find the values of x that make the equation true when y=0.
  • We can expand the equation, set y=0, factor the quadratic equation, solve for x, and find the corresponding values of y.
  • The y-intercept is an essential concept in algebra, and it is used in a variety of applications.

References

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Introduction

In our previous article, we determined the y-intercept of the given equation (x-5)(x+2)=y. In this article, we will answer some frequently asked questions related to the y-intercept of quadratic equations.

Q&A

Q: What is the y-intercept of a quadratic equation?

A: The y-intercept of a quadratic equation is the point at which the graph of the equation intersects the y-axis. It is a crucial concept in understanding the behavior of functions and their graphs.

Q: How do I find the y-intercept of a quadratic equation?

A: To find the y-intercept of a quadratic equation, you need to find the values of x that make the equation true when y=0. You can do this by expanding the equation, setting y=0, factoring the quadratic equation, solving for x, and finding the corresponding values of y.

Q: What is the difference between the x-intercept and the y-intercept?

A: The x-intercept of a quadratic equation is the point at which the graph of the equation intersects the x-axis, while the y-intercept is the point at which the graph of the equation intersects the y-axis.

Q: Can the y-intercept of a quadratic equation be negative?

A: Yes, the y-intercept of a quadratic equation can be negative. In fact, the y-intercept can be any real number.

Q: How do I determine the y-intercept of a quadratic equation with a negative leading coefficient?

A: To determine the y-intercept of a quadratic equation with a negative leading coefficient, you can follow the same steps as before: expand the equation, set y=0, factor the quadratic equation, solve for x, and find the corresponding values of y.

Q: Can the y-intercept of a quadratic equation be a fraction?

A: Yes, the y-intercept of a quadratic equation can be a fraction. In fact, the y-intercept can be any real number, including fractions.

Q: How do I graph a quadratic equation with a negative leading coefficient?

A: To graph a quadratic equation with a negative leading coefficient, you can use the y-intercept and the x-intercepts to plot the graph. You can also use the vertex form of the quadratic equation to graph the function.

Q: Can the y-intercept of a quadratic equation be a complex number?

A: Yes, the y-intercept of a quadratic equation can be a complex number. In fact, the y-intercept can be any complex number.

Q: How do I determine the y-intercept of a quadratic equation with a complex leading coefficient?

A: To determine the y-intercept of a quadratic equation with a complex leading coefficient, you can follow the same steps as before: expand the equation, set y=0, factor the quadratic equation, solve for x, and find the corresponding values of y.

Conclusion

In conclusion, the y-intercept of a quadratic equation is an essential concept in understanding the behavior of functions and their graphs. We have answered some frequently asked questions related to the y-intercept of quadratic equations, and we hope that this article has been helpful in clarifying any doubts you may have had.

Key Takeaways

  • The y-intercept of a quadratic equation is the point at which the graph of the equation intersects the y-axis.
  • To find the y-intercept, you need to find the values of x that make the equation true when y=0.
  • The y-intercept can be any real number, including fractions and complex numbers.
  • The y-intercept is an essential concept in understanding the behavior of functions and their graphs.

References

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