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To find the quadratic equation with roots \( -5 \) and \( 1 \), and with a leading coefficient of \( 5 \), we start with the general form of a quadratic equation given the roots:

\[
a(x - r_1)(x - r_2) = 0
\]

where \( r_1 \) and \( r_2 \) are the roots, and \( a \) is the leading coefficient.

Here, the roots are \( -5 \) and \( 1 \), and the leading coefficient \( a \) is \( 5 \). Plugging these values in gives:

\[
5(x - (-5))(x - 1) = 0
\]

This simplifies to:

\[
5(x + 5)(x - 1) = 0
\]

Now, expand the expression:

\[
5[(x)(x) + (x)(-1) + (5)(x) + (5)(-1)] = 0
\]

\[
5(x^2 + 4x - 5) = 0
\]

Thus, the quadratic equation is:

\[
5x^2 + 20x - 25 = 0
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you find a quadratic equation when only given the sum and product of the roots?
2. What would the quadratic equation be if the leading coefficient was 1 instead of 5?
3. How does changing the leading coefficient affect the shape of a parabola?
4. Can you derive the quadratic equation from a graph where the roots are known?
5. How do you factor a quadratic equation like \( 5x^2 + 20x - 25 \)?

**Tip:** When given roots of a quadratic equation, remember that the general form is \( a(x - r_1)(x - r_2) \), where \( a \) can be any non-zero constant, and \( r_1 \) and \( r_2 \) are the roots.

Write the quadratic equation whose roots are -5 and 1, and whose leading coefficient is 5.

Learn how to write the quadratic equation with roots -5 and 1, and a leading coefficient of 5. This step-by-step solution covers essential algebraic concepts such as factoring and expanding quadratic expressions.

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