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Question 6:

You are asked to analyze the graph of the quadratic function $y = ax^2 + bx + c$, and determine whether each expression in the table is positive, negative, or zero based on the graph.

• $b^2 - 4ac$: This is the discriminant of the quadratic equation, and it helps determine the nature of the roots of the equation.

  • If $b^2 - 4ac > 0$, the equation has two distinct real roots.
  • If $b^2 - 4ac = 0$, the equation has one real root (repeated or double root).
  • If $b^2 - 4ac < 0$, the equation has no real roots (complex roots).

• $b$: This refers to the linear coefficient of the quadratic equation.

From the graph, we see that the parabola touches the x-axis once, which means the quadratic has equal roots. Therefore:

• $b^2 - 4ac = 0$ (discriminant is zero because the equation has equal roots).
• You need to determine the sign of $b$ based on the orientation of the graph and the values of the vertex.

Fill out the table based on the information from the graph.

Question 7:

You need to find the exact values of $k$ for which the equation $kx^2 + (k+1)x + 1 = 0$ has equal roots.

For the quadratic equation $ax^2 + bx + c = 0$ to have equal roots, its discriminant must be equal to 0: $b^2 - 4ac = 0$ In your equation $kx^2 + (k+1)x + 1 = 0$, the coefficients are:

• $a = k$
• $b = k+1$
• $c = 1$

Now, apply the discriminant condition: $(k+1)^2 - 4(k)(1) = 0$

Expanding and solving: $(k+1)^2 - 4k = 0$ $k^2 + 2k + 1 - 4k = 0$ $k^2 - 2k + 1 = 0$ This simplifies to: $(k - 1)^2 = 0$ Thus, $k = 1$.

Answer:

The exact value of $k$ for which the equation has equal roots is $k = 1$.

Would you like further details or clarifications? Here are 5 related questions to expand your understanding:

1. What is the geometric interpretation of the discriminant in quadratic equations?
2. How does the discriminant change the number of real solutions to a quadratic equation?
3. Can the discriminant be applied to other types of polynomials?
4. How does changing the value of $a$ in a quadratic equation affect the graph of the parabola?
5. What is the significance of a "double root" in quadratic equations?

Tip: When solving quadratic equations, always check the discriminant first to determine the nature of the roots.