find the roots of (x^2)+3x-28=0. hence, from the quadratic equation such that its roots are 1/3 of the roots of the equation (x^2)+3x-28=0

To solve this problem, we'll go through it step by step. First, we'll find the roots of the quadratic equation \(x^2 + 3x - 28 = 0\). Then, we'll determine the quadratic equation whose roots are \( \frac{1}{3} \) of the roots of this original equation.

### Step 1: Find the roots of \(x^2 + 3x - 28 = 0\)
This is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 3\), and \(c = -28\).

The roots of the quadratic equation can be found using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values of \(a\), \(b\), and \(c\):

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-28)}}{2(1)}
\]
\[
x = \frac{-3 \pm \sqrt{9 + 112}}{2}
\]
\[
x = \frac{-3 \pm \sqrt{121}}{2}
\]
\[
x = \frac{-3 \pm 11}{2}
\]

So, the roots are:

\[
x_1 = \frac{-3 + 11}{2} = \frac{8}{2} = 4
\]

\[
x_2 = \frac{-3 - 11}{2} = \frac{-14}{2} = -7
\]

### Step 2: Find the quadratic equation whose roots are \( \frac{1}{3} \) of the roots of the original equation

The original roots are 4 and -7. We need to find the quadratic equation whose roots are:

\[
\frac{4}{3} \quad \text{and} \quad \frac{-7}{3}
\]

If the roots of a quadratic equation are \( \alpha \) and \( \beta \), the quadratic equation can be expressed as:

\[
x^2 - (\alpha + \beta)x + \alpha\beta = 0
\]

Here, \( \alpha = \frac{4}{3} \) and \( \beta = \frac{-7}{3} \).

First, calculate \( \alpha + \beta \):

\[
\alpha + \beta = \frac{4}{3} + \frac{-7}{3} = \frac{4 - 7}{3} = \frac{-3}{3} = -1
\]

Now, calculate \( \alpha \beta \):

\[
\alpha \beta = \frac{4}{3} \times \frac{-7}{3} = \frac{-28}{9}
\]

So, the quadratic equation is:

\[
x^2 - (-1)x + \frac{-28}{9} = 0
\]

Simplifying this:

\[
x^2 + x - \frac{28}{9} = 0
\]

To express this equation in a standard form (with integer coefficients), multiply the entire equation by 9:

\[
9x^2 + 9x - 28 = 0
\]

This is the required quadratic equation whose roots are \( \frac{1}{3} \) of the roots of the original equation \(x^2 + 3x - 28 = 0\).

### Summary
- The roots of the equation \(x^2 + 3x - 28 = 0\) are \(4\) and \(-7\).
- The quadratic equation with roots \( \frac{1}{3} \) of these roots is \(9x^2 + 9x - 28 = 0\).

Would you like more details on any part of the solution, or do you have any questions?

### Related Questions:
1. What is the relationship between the discriminant and the nature of the roots of a quadratic equation?
2. How can you derive the quadratic formula from completing the square?
3. What are the sum and product of the roots for any quadratic equation?
4. How do you solve a quadratic equation using the factorization method?
5. What are the conditions for a quadratic equation to have real and distinct roots?

**Tip:** For any quadratic equation, the sum and product of its roots can give quick insights into the properties of the equation without fully solving it.

Learn how to find the roots of the quadratic equation x^2 + 3x - 28, and how to derive a new quadratic equation whose roots are 1/3 of the original roots. This detailed solution covers essential algebraic concepts and the application of the quadratic formula. Ideal for students in Grades 9-10.

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