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The questions seem to be related to quadratic equations. Here's what the image asks:

Question 1:

Work out the discriminant, hence state the number of solutions (roots) to the equation:

$x^2 - 7x + 12 = 0$

Solution 1:

The discriminant ($\Delta$) for a quadratic equation $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

For the given equation $x^2 - 7x + 12 = 0$, we have:

• $a = 1$
• $b = -7$
• $c = 12$

Now, calculate the discriminant:

$\Delta = (-7)^2 - 4(1)(12) = 49 - 48 = 1$

Since the discriminant is positive and equal to 1, the quadratic equation has two real and distinct roots.

Question 2:

Given the graphs, determine the number of roots:

The graphs show two different parabolas. Based on the visual appearance:

1. First graph: The parabola intersects the x-axis in two points, so there are two real roots.
2. Second graph: The parabola touches the x-axis at only one point (vertex), indicating a double root or one real root.

Question 3:

If all the values of a such that $x^2 + 6x + a = 0$ have no real solutions, solve for $a$:

For a quadratic equation to have no real solutions, its discriminant must be negative.

The discriminant for $x^2 + 6x + a = 0$ is:

$\Delta = 6^2 - 4(1)(a) = 36 - 4a$

For no real solutions:

$36 - 4a < 0$

Solving for $a$:

$36 < 4a$ $a > 9$

Thus, for the equation to have no real solutions, $a$ must be greater than 9.

Question 4:

Solve by completing the square:

The question asks to solve a quadratic equation by completing the square. Since the specific equation is not clear in the image, let me know if you'd like to proceed with an example or clarification for this part.

Do you want further explanation on any of these parts, or do you have any questions?

Follow-up Questions:

1. Can you explain how the discriminant affects the nature of the roots?
2. What is the method for solving quadratic equations by completing the square?
3. How does the graph of a quadratic equation help in determining the number of real roots?
4. Can a quadratic equation have no real roots but still have complex solutions?
5. What is the significance of the vertex of a parabola in determining the number of roots?

Tip:

When solving quadratic equations, always check the discriminant first to quickly determine how many and what type of solutions you will have.