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To find the zeros of the quadratic polynomial \(x^2 + 7x + 10\), we'll use the quadratic formula, which is:

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

Here, \(a = 1\), \(b = 7\), and \(c = 10\).

Plugging these values into the formula gives:

\[
x = \frac{{-7 \pm \sqrt{{7^2 - 4 \times 1 \times 10}}}}{2 \times 1}
\]

\[
x = \frac{{-7 \pm \sqrt{{49 - 40}}}}{2}
\]

\[
x = \frac{{-7 \pm \sqrt{9}}}{2}
\]

\[
x = \frac{{-7 \pm 3}}{2}
\]

This gives two solutions:

\[
x = \frac{{-7 + 3}}{2} = -2
\]

\[
x = \frac{{-7 - 3}}{2} = -5
\]

So the zeros of the polynomial are \(x = -2\) and \(x = -5\).

To verify the relationship between the zeros and the coefficients:
- The sum of the zeros (\(-2\) and \(-5\)) is \(-2 + (-5) = -7\), which is equal to \(-b/a\) (as \(b = 7\), \(-b = -7\)).
- The product of the zeros (\(-2\) \(\times\) \(-5\)) is \(10\), which is equal to \(c/a\) (as \(c = 10\)).

These results verify the relationships specified by Vieta's formulas, demonstrating that the sum of the zeros equals the negative of the coefficient of \(x\) divided by the leading coefficient, and the product of the zeros equals the constant term divided by the leading coefficient.

Do you need further details or have any questions on this topic?

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Here are some related questions you might consider:
1. How would you factor the quadratic \(x^2 + 7x + 10\) into linear factors?
2. What would change in the calculation if the coefficients were different?
3. Can you apply the quadratic formula to find the roots of \(x^2 - 3x + 2\)?
4. How does the discriminant in the quadratic formula influence the nature of the roots?
5. What are Vieta’s formulas and how do they relate the roots and coefficients of polynomials?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first; it tells you about the nature of the roots (real and distinct, real and equal, or complex).

Find the zeroes of the quadratic polynomial x^2 + 7x + 10 and verify the relationship between the zeroes and the coefficients.

Learn how to find the zeroes of the quadratic polynomial x^2 + 7x + 10 using the quadratic formula, and verify the relationship between the zeroes and the coefficients as explained by Vieta's formulas. This step-by-step solution is suitable for students in Grades 9-10.

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