Given The Graph Of F ( X ) = X 2 + 6 X + 8 F(x)=x^2+6x+8 F ( X ) = X 2 + 6 X + 8 , Select ALL True Statements About The Solutions:A. -4 Is A Solution Because It Is An X X X -intercept.B. -3 Is A Solution Because It Is The X X X -value Of The Vertex.C. -1 Is A Solution Because

Solving Quadratic Equations: Understanding the Graph of $f(x)=x^2+6x+8$

Quadratic equations are a fundamental concept in mathematics, and understanding their graphs is crucial for solving various problems in algebra and calculus. In this article, we will focus on the graph of the quadratic function $f(x)=x^2+6x+8$ and examine the truth of several statements about its solutions.

The Graph of $f(x)=x^2+6x+8$

To begin with, let's analyze the graph of $f(x)=x^2+6x+8$. This quadratic function can be written in the form $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. By completing the square, we can rewrite the function as:

$$f(x)=(x+3)^2-1$$

This form reveals that the vertex of the parabola is at $(-3,-1)$.

Understanding $x$-Intercepts

An $x$-intercept is a point on the graph where the function intersects the $x$-axis. In other words, it is a point where $f(x)=0$. To find the $x$-intercepts of the graph of $f(x)=x^2+6x+8$, we need to solve the equation $x^2+6x+8=0$.

Using the quadratic formula, we can find the solutions to this equation:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

where $a=1$, $b=6$, and $c=8$. Plugging in these values, we get:

$$x=\frac{-6\pm\sqrt{6^2-4(1)(8)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x=\frac{-6\pm\sqrt{36-32}}{2}$$

$$x=\frac{-6\pm\sqrt{4}}{2}$$

$$x=\frac{-6\pm2}{2}$$

Therefore, the solutions to the equation $x^2+6x+8=0$ are:

$$x=\frac{-6+2}{2}=-2$$

$$x=\frac{-6-2}{2}=-4$$

These two values represent the $x$-intercepts of the graph of $f(x)=x^2+6x+8$.

Evaluating Statement A

Now that we have found the $x$-intercepts of the graph of $f(x)=x^2+6x+8$, let's examine statement A: "-4 is a solution because it is an $x$-intercept."

As we have just shown, -4 is indeed an $x$-intercept of the graph of $f(x)=x^2+6x+8$. Therefore, statement A is TRUE.

Evaluating Statement B

Next, let's examine statement B: "-3 is a solution because it is the $x$-value of the vertex."

As we have already discussed, the vertex of the parabola is at $(-3,-1)$. However, this does not mean that -3 is a solution to the equation $x^2+6x+8=0$. In fact, we can plug -3 into the equation to verify that it is not a solution:

$$(-3)^2+6(-3)+8=9-18+8=-1$$

Since -1 is not equal to 0, -3 is not a solution to the equation. Therefore, statement B is FALSE.

Evaluating Statement C

Finally, let's examine statement C: "-1 is a solution because [insert reason here]."

Unfortunately, statement C is incomplete, and we cannot determine its truth value without more information. However, we can try to find the solution to the equation $x^2+6x+8=0$ using the quadratic formula:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

where $a=1$, $b=6$, and $c=8$. Plugging in these values, we get:

$$x=\frac{-6\pm\sqrt{6^2-4(1)(8)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x=\frac{-6\pm\sqrt{36-32}}{2}$$

$$x=\frac{-6\pm\sqrt{4}}{2}$$

$$x=\frac{-6\pm2}{2}$$

Therefore, the solutions to the equation $x^2+6x+8=0$ are:

$$x=\frac{-6+2}{2}=-2$$

$$x=\frac{-6-2}{2}=-4$$

As we can see, -1 is not a solution to the equation. Therefore, statement C is FALSE.

Conclusion

In conclusion, we have examined three statements about the solutions to the equation $x^2+6x+8=0$. We found that statement A is TRUE, statement B is FALSE, and statement C is FALSE. By understanding the graph of the quadratic function $f(x)=x^2+6x+8$, we can determine the truth value of these statements and gain a deeper understanding of quadratic equations.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Purplemath
• [3] "Solving Quadratic Equations" by Khan Academy
  Quadratic Equations: A Q&A Guide

Quadratic equations are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra and calculus. In this article, we will provide a comprehensive Q&A guide to help you better understand quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, you can set each binomial equal to zero and solve for x.
• Quadratic formula: The quadratic formula is a general method for solving quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a.
• Graphing: You can graph the quadratic function and find the x-intercepts, which represent the solutions to the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a general method for solving quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by b^2 - 4ac. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the discriminant. If the discriminant is:

• Positive, the equation has two distinct real solutions.
• Zero, the equation has one real solution.
• Negative, the equation has no real solutions.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is given by the coordinates (h, k), where h = -b / 2a and k = f(h).

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you can use the formula h = -b / 2a and k = f(h). Alternatively, you can graph the function and find the vertex by inspection.

Q: What is the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex of a quadratic function. It is given by the equation x = h, where h is the x-coordinate of the vertex.

Q: How do I find the axis of symmetry?

A: To find the axis of symmetry, you can use the formula x = h, where h is the x-coordinate of the vertex. Alternatively, you can graph the function and find the axis of symmetry by inspection.

Q: What is the difference between a quadratic equation and a quadratic function?

A: A quadratic equation is a polynomial equation of degree two, while a quadratic function is a polynomial function of degree two. A quadratic equation is typically written in the form ax^2 + bx + c = 0, while a quadratic function is typically written in the form f(x) = ax^2 + bx + c.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra and calculus. This Q&A guide provides a comprehensive overview of quadratic equations, including how to solve them, the quadratic formula, the discriminant, the vertex, and the axis of symmetry. We hope this guide has been helpful in your understanding of quadratic equations.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Purplemath
• [3] "Solving Quadratic Equations" by Khan Academy