Transform  −3x(5+2x)=−17−3*x*(5+2*x*)=−17  into  ax2+bx+c=0*ax*2+*bx*+*c*=0  form.

To transform the given equation \( -3x(5 + 2x) = -17 \) into the standard quadratic form \( ax^2 + bx + c = 0 \), let's follow these steps:

### Step 1: Distribute \(-3x\) on the left-hand side
\[
-3x(5 + 2x) = -3x \cdot 5 - 3x \cdot 2x
\]
\[
= -15x - 6x^2
\]

### Step 2: Substitute the expanded terms into the equation
Substituting back into the original equation, we have:
\[
-15x - 6x^2 = -17
\]

### Step 3: Rearrange the equation to set it to zero
To convert it to the standard quadratic form, \( ax^2 + bx + c = 0 \), move all terms to one side:
\[
-6x^2 - 15x + 17 = 0
\]

### Step 4: Multiply through by -1 to have the leading coefficient positive
\[
6x^2 + 15x - 17 = 0
\]

Now, we have the equation in the desired quadratic form:

\[
ax^2 + bx + c = 0
\]

Where:

- \( a = 6 \)
- \( b = 15 \)
- \( c = -17 \)

### Final Equation:
\[
6x^2 + 15x - 17 = 0
\]

Would you like more details on how to solve this quadratic equation or any other questions?

### Relative Questions:
1. How do you find the roots of the quadratic equation \( 6x^2 + 15x - 17 = 0 \) using the quadratic formula?
2. Can the quadratic equation \( 6x^2 + 15x - 17 = 0 \) be factored, or is the quadratic formula necessary?
3. What does the discriminant tell us about the nature of the roots for \( 6x^2 + 15x - 17 = 0 \)?
4. How do the coefficients \( a = 6 \), \( b = 15 \), and \( c = -17 \) affect the graph of the quadratic equation?
5. What is the vertex of the parabola represented by the equation \( 6x^2 + 15x - 17 = 0 \)?

### Tip:
Always check the discriminant \(\Delta = b^2 - 4ac\) before attempting to solve a quadratic equation, as it tells you the nature and number of roots!

Learn how to transform the equation -3x(5 + 2x) = -17 into the standard quadratic form ax^2 + bx + c = 0. Understand the steps involved in expanding and rearranging the equation. Suitable for Grades 7-9, covering algebraic concepts and quadratic equations.

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