Find a quadratic equation that satisfies the given condition: The roots are 3 greater than the roots of x^2+3x-10=0

We are tasked with finding a quadratic equation whose roots are 3 greater than the roots of the quadratic equation:

\[
x^2 + 3x - 10 = 0
\]

### Step 1: Find the roots of the given quadratic equation.

Using the quadratic formula:

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

For the equation \( x^2 + 3x - 10 = 0 \), we have:
- \( a = 1 \)
- \( b = 3 \)
- \( c = -10 \)

Substituting into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-10)}}{2(1)}
\]
\[
x = \frac{-3 \pm \sqrt{9 + 40}}{2}
\]
\[
x = \frac{-3 \pm \sqrt{49}}{2}
\]
\[
x = \frac{-3 \pm 7}{2}
\]

Thus, the roots are:

\[
x = \frac{-3 + 7}{2} = 2 \quad \text{and} \quad x = \frac{-3 - 7}{2} = -5
\]

### Step 2: Find the new roots.

We are asked to find a quadratic equation whose roots are 3 greater than the original roots. So, the new roots are:

\[
2 + 3 = 5 \quad \text{and} \quad -5 + 3 = -2
\]

### Step 3: Form the new quadratic equation.

The quadratic equation with roots \(5\) and \(-2\) is given by:

\[
(x - 5)(x + 2) = 0
\]

Expanding this:

\[
x^2 + 2x - 5x - 10 = 0
\]
\[
x^2 - 3x - 10 = 0
\]

Thus, the quadratic equation is:

\[
x^2 - 3x - 10 = 0
\]

This is the quadratic equation whose roots are 3 greater than the roots of \(x^2 + 3x - 10 = 0\).

---

Would you like further details or have any questions?

### Related Questions:
1. What is the discriminant of the quadratic equation \(x^2 - 3x - 10 = 0\)?
2. How do you derive the quadratic formula?
3. How do you find the sum and product of the roots of a quadratic equation?
4. How do you solve a quadratic equation by factoring?
5. Can quadratic equations have complex roots?

### Tip:
When finding roots, always double-check your signs in the quadratic formula to avoid errors.

Explore how to solve for a quadratic equation whose roots are 3 greater than the roots of x² + 3x - 10. This step-by-step guide walks through the use of the quadratic formula and root-shifting techniques. Suitable for students in Grades 9-11.

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