14. Draw a rough diagram for the quadratic polynomial $$ a x^{2}+b x+c,(a
eq 0) $$ when $$ \mathrm{a}0 $$. 15. Find the value of k for the quadratic equatioin $$ k x(x-2)+6=0 $$. So that it has two equal roots.

Question

14. Draw a rough diagram for the quadratic polynomial $$ a x^{2}+b x+c,(a 
eq 0) $$ when $$ \mathrm{a}>0 $$. 15. Find the value of k for the quadratic equatioin $$ k x(x-2)+6=0 $$. So that it has two equal roots.

UpStudy AI · Accepted Answer

**14. Rough Diagram for $$ a x^2 + b x + c $$ when $$ a>0 $$:**

Since $$ a>0 $$, the parabola opens upward. A rough sketch of the graph might look like:

```
          |
        V |       •
          |      • •
          |     •   •
          |----•-----•----> x
          |   •       •
          |  •         •
          | •           •
          |•             •
          |
```

Here, the point labeled $$ V $$ represents the vertex of the parabola, and the vertical line passing through it is the axis of symmetry.

---

**15. Finding the value of $$ k $$ for which the quadratic equation $$ kx(x-2)+6=0 $$ has two equal roots:**

1. **Write the equation in standard quadratic form:**

Expand the given equation:
   $$
   kx(x-2)+6 = kx^2 - 2kx + 6 = 0.
   $$
   Here, the quadratic is of the form $$ Ax^2 + Bx + C = 0 $$ with:
   $$
   A = k,\quad B = -2k,\quad C = 6.
   $$

2. **Set the discriminant equal to zero for equal roots:**

For a quadratic equation to have equal (repeated) roots, the discriminant must be zero:
   $$
   \Delta = B^2 - 4AC = 0.
   $$
   Substitute the values of $$ A $$, $$ B $$, and $$ C $$:
   $$
   (-2k)^2 - 4(k)(6) = 0.
   $$
   Simplify:
   $$
   4k^2 - 24k = 0.
   $$

3. **Solve for $$ k $$:**

Factor the equation:
   $$
   4k(k-6) = 0.
   $$
   Therefore, the solutions are:
   $$
   k=0\quad \text{or}\quad k=6.
   $$

4. **Exclude any non-quadratic case:**

If $$ k=0 $$, the equation becomes a constant equation $$ 6 = 0 $$, which is invalid. Hence, the only valid solution for the quadratic is:
   $$
   k=6.
   $$

---

**Answer:**  
For the upward opening parabola, the rough diagram is as shown above.  
The value of $$ k $$ that makes the quadratic have two equal roots is $$ k=6 $$.
**14. Rough Diagram for $$ a x^2 + b x + c $$ when $$ a>0 $$:** The parabola opens upward with its vertex at the lowest point. **15. Value of $$ k $$ for the equation $$ kx(x-2)+6=0 $$ to have two equal roots:** Set the discriminant to zero: $$ (-2k)^2 - 4(k)(6) = 0 $$. Solving this gives $$ k = 6 $$. **Answer:** The parabola opens upward with its vertex at the lowest point. The value of $$ k $$ is 6.

Draw a rough diagram for the quadratic polynomial when .

Find the value of k for the quadratic equatioin . So that it has two equal
roots.

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Draw a rough diagram for the quadratic polynomial when . Find the value of k for the quadratic equatioin . So that it has two equal roots.