Prove That The Roots Of X 2 + ( 1 + K ) X + K − 3 = 0 X^2 + (1+k)x + K - 3 = 0 X 2 + ( 1 + K ) X + K − 3 = 0 Are Real For All Values Of K K K .2. The Graph Of F ( X ) = − X 2 + 8 X + 20 F(x) = -x^2 + 8x + 20 F ( X ) = − X 2 + 8 X + 20 Is Sketched Below. The Graph Intersects The X-axis At ( − 2 , 0 (-2, 0 ( − 2 , 0 ] And ( 10 , 0 (10, 0 ( 10 , 0 ], And

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1.1 Introduction

In this section, we will prove that the roots of the quadratic equation $x^2 + (1+k)x + k - 3 = 0$ are real for all values of $k$. To do this, we will use the discriminant of the quadratic equation, which is given by the formula $b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

1.2 The Discriminant

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by the formula $b^2 - 4ac$. If the discriminant is positive, then the roots of the quadratic equation are real and distinct. If the discriminant is zero, then the roots of the quadratic equation are real and equal. If the discriminant is negative, then the roots of the quadratic equation are complex.

1.3 Proving the Roots are Real

To prove that the roots of the quadratic equation $x^2 + (1+k)x + k - 3 = 0$ are real for all values of $k$, we need to show that the discriminant is non-negative for all values of $k$. The discriminant of the quadratic equation is given by the formula $(1+k)^2 - 4(1)(k-3)$.

1.4 Simplifying the Discriminant

We can simplify the discriminant by expanding the square and combining like terms. The discriminant is given by the formula $(1+k)^2 - 4(1)(k-3) = 1 + 2k + k^2 - 4k + 12 = k^2 - 2k + 13$.

1.5 Proving the Discriminant is Non-Negative

To prove that the discriminant is non-negative for all values of $k$, we can complete the square on the quadratic expression $k^2 - 2k + 13$. We can write the quadratic expression as $(k-1)^2 + 12$. Since the square of a real number is always non-negative, we know that $(k-1)^2 \geq 0$ for all values of $k$. Therefore, we have $(k-1)^2 + 12 \geq 12$ for all values of $k$.

1.6 Conclusion

We have shown that the discriminant of the quadratic equation $x^2 + (1+k)x + k - 3 = 0$ is non-negative for all values of $k$. Therefore, we can conclude that the roots of the quadratic equation are real for all values of $k$.

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2.1 Introduction

In this section, we will analyze the graph of the quadratic function $f(x) = -x^2 + 8x + 20$. The graph of a quadratic function is a parabola that opens upward or downward. The vertex of the parabola is the point where the graph changes direction.

2.2 Finding the Vertex

To find the vertex of the parabola, we can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In this case, we have $a = -1$ and $b = 8$. Plugging these values into the formula, we get $x = -\frac{8}{2(-1)} = 4$.

2.3 Finding the y-Coordinate of the Vertex

To find the y-coordinate of the vertex, we can plug the x-coordinate into the quadratic function. We have $f(4) = -(4)^2 + 8(4) + 20 = -16 + 32 + 20 = 36$.

2.4 Writing the Vertex Form of the Quadratic Function

We can write the vertex form of the quadratic function as $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. In this case, we have $a = -1$, $h = 4$, and $k = 36$. Plugging these values into the formula, we get $f(x) = -(x-4)^2 + 36$.

2.5 Finding the x-Intercepts

To find the x-intercepts of the parabola, we can set the quadratic function equal to zero and solve for $x$. We have $-(x-4)^2 + 36 = 0$. Expanding the square and combining like terms, we get $-x^2 + 8x - 16 + 36 = 0$. Simplifying the equation, we get $-x^2 + 8x + 20 = 0$.

2.6 Solving the Quadratic Equation

To solve the quadratic equation $-x^2 + 8x + 20 = 0$, we can use the quadratic formula. The quadratic formula is given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation. In this case, we have $a = -1$, $b = 8$, and $c = 20$. Plugging these values into the formula, we get $x = \frac{-8 \pm \sqrt{8^2 - 4(-1)(20)}}{2(-1)} = \frac{-8 \pm \sqrt{64 + 80}}{-2} = \frac{-8 \pm \sqrt{144}}{-2} = \frac{-8 \pm 12}{-2}$.

2.7 Finding the x-Intercepts

We have two possible solutions for the x-intercepts: $x = \frac{-8 + 12}{-2} = \frac{4}{-2} = -2$ and $x = \frac{-8 - 12}{-2} = \frac{-20}{-2} = 10$.

2.8 Conclusion

We have analyzed the graph of the quadratic function $f(x) = -x^2 + 8x + 20$. We have found the vertex of the parabola, written the vertex form of the quadratic function, and found the x-intercepts of the parabola.

3.1 Introduction

In this section, we will answer some common questions about quadratic equations and graphs. Quadratic equations are a type of polynomial equation that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Quadratic graphs are the graphs of quadratic equations and can be used to model a wide range of real-world phenomena.

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

A: A quadratic equation is a mathematical expression that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. A quadratic graph, on the other hand, is the visual representation of a quadratic equation on a coordinate plane.

3.3 Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you need to look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

3.4 Q: How do I find the vertex of the parabola?

A: To find the vertex of the parabola, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic equation. Once you have the x-coordinate of the vertex, you can plug it into the quadratic equation to find the y-coordinate.

3.5 Q: How do I find the x-intercepts of the parabola?

A: To find the x-intercepts of the parabola, you can set the quadratic equation equal to zero and solve for $x$. You can use the quadratic formula to solve for $x$.

3.6 Q: What is the significance of the discriminant in a quadratic equation?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation. It can be used to determine the nature of the roots of the equation. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

3.7 Q: How do I determine if a quadratic equation has real or complex roots?

A: To determine if a quadratic equation has real or complex roots, you can calculate the discriminant. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation can be written in the form $ax^2 + bx + c = 0$, while a linear equation can be written in the form $ax + b = 0$.

3.9 Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula. The quadratic formula is given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

3.10 Q: What is the significance of the vertex form of a quadratic equation?

A: The vertex form of a quadratic equation is a way of writing the equation in a form that makes it easier to graph. It is given by the formula $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola.

3.11 Conclusion

In this section, we have answered some common questions about quadratic equations and graphs. We have discussed the difference between a quadratic equation and a quadratic graph, how to determine the direction of the parabola, how to find the vertex of the parabola, and how to find the x-intercepts of the parabola. We have also discussed the significance of the discriminant, how to determine if a quadratic equation has real or complex roots, and the difference between a quadratic equation and a linear equation.