Let \[x=h\] and \[x=m\] be solutions to the given equation, with \[h>m\]. What is the value of \[h-m\]x^2-10x+21=0

The given quadratic equation is:

\[
h - m x^2 - 10x + 21 = 0
\]

We are told that \(x = h\) and \(x = m\) are the solutions to this equation, and \(h > m\). Using the property of quadratic equations, the standard form can be written as:

\[
a x^2 + bx + c = 0
\]

where the solutions are \(h\) and \(m\). We can use Vieta's formulas to relate the coefficients of the quadratic equation to the roots \(h\) and \(m\):

1. The sum of the roots (from Vieta’s relations) is:

\[
h + m = -\frac{b}{a}
\]

2. The product of the roots is:

\[
h \cdot m = \frac{c}{a}
\]

Comparing the given quadratic equation with the standard form, we have:

\[
h - m = 1, \quad b = -10, \quad c = 21
\]

Therefore:

1. From the sum of the roots: \(h + m = -\frac{-10}{1} = 10\).
2. From the product of the roots: \(h \cdot m = \frac{21}{1} = 21\).

Now, we have the system of equations:

\[
h + m = 10 \quad \text{(1)}
\]
\[
h \cdot m = 21 \quad \text{(2)}
\]

To solve this system, substitute \(h = 10 - m\) from equation (1) into equation (2):

\[
(10 - m) \cdot m = 21
\]

Expanding this:

\[
10m - m^2 = 21
\]

Rearrange into a standard quadratic form:

\[
m^2 - 10m + 21 = 0
\]

Now, solve for \(m\) using the quadratic formula:

\[
m = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(21)}}{2(1)}
\]
\[
m = \frac{10 \pm \sqrt{100 - 84}}{2}
\]
\[
m = \frac{10 \pm \sqrt{16}}{2}
\]
\[
m = \frac{10 \pm 4}{2}
\]

Thus, we have two solutions:

\[
m = \frac{10 + 4}{2} = 7 \quad \text{or} \quad m = \frac{10 - 4}{2} = 3
\]

Since \(h > m\), we choose \(m = 3\), which gives \(h = 10 - 3 = 7\).

Finally, the value of \(h - m = 7 - 3 = 4\).

So, the value of \(h - m\) is \(4\).

Do you have any questions or need more details? 

Here are five related questions to further explore this topic:

1. What are Vieta's formulas and how do they apply to quadratic equations?
2. How can the quadratic formula be derived?
3. What are the steps to solve a quadratic equation by factoring?
4. How does the discriminant determine the nature of the roots in a quadratic equation?
5. What are real-life applications of quadratic equations?

**Tip**: When solving quadratic equations, always check if factoring is possible before using the quadratic formula for simplicity.

Let [x = h] and [x = m] be solutions to the given equation, with [h > m]. What is the value of [h - m] for the equation x^2 - 10x + 21 = 0?

Learn how to solve the quadratic equation x^2 - 10x + 21 using Vieta's formulas. This problem covers important algebraic concepts, including the sum and product of roots, the quadratic formula, and applying Vieta's relations. Suitable for students in Grades 8-10.

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