If a + b = 10 and ab = 21, what are the values of a and b?

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Question: If a + b = 10 and ab = 21, what are the values of a and b?

Options:

Correct Answer: 3 and 7

Solution:

The roots of the equation x^2 - 10x + 21 = 0 are found using the quadratic formula, yielding a = 3 and b = 7.

If a + b = 10 and ab = 21, what are the values of a and b?

If a + b = 10 and ab = 21, what are the values of a and b?

Step 1: Start with the equations given: a + b = 10 and ab = 21.
Step 2: Rewrite the first equation (a + b = 10) in a different form: b = 10 - a.
Step 3: Substitute b in the second equation (ab = 21) with (10 - a): a(10 - a) = 21.
Step 4: Expand the equation: 10a - a^2 = 21.
Step 5: Rearrange the equation to standard quadratic form: -a^2 + 10a - 21 = 0.
Step 6: Multiply the entire equation by -1 to make the leading coefficient positive: a^2 - 10a + 21 = 0.
Step 7: Now, use the quadratic formula: a = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = -10, and c = 21.
Step 8: Calculate the discriminant: (-10)² - 4(1)(21) = 100 - 84 = 16.
Step 9: Find the square root of the discriminant: √16 = 4.
Step 10: Substitute back into the quadratic formula: a = (10 ± 4) / 2.
Step 11: Calculate the two possible values for a: a = (10 + 4) / 2 = 7 and a = (10 - 4) / 2 = 3.
Step 12: Therefore, the values of a and b are a = 3 and b = 7.

Quadratic Equations – The question tests the ability to form and solve a quadratic equation based on given sums and products of roots.
Roots of Equations – It assesses understanding of finding roots using the quadratic formula.