Post A 100-300 Word Response To Each Question

Post A 100 300 Word Response To Each Individual Bulleted Question

Post a word response to each individual (bulleted) question. Responses should be submitted as individual responses, not in an essay format. What is the quadratic formula? What is it used for? Give an example of a quadratic equation and solve it using the quadratic formula. (Round answers to the nearest hundredth) What is the formula for the discriminant in a quadratic equation? How can you use it to determine the number of solutions to a quadratic equation?

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The quadratic formula is a mathematical expression used to find the solutions, or roots, of a quadratic equation, which typically takes the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The formula provides a direct method to determine the values of x that satisfy the equation, especially when factoring is difficult or impossible. The quadratic formula is written as:

x = (-b ± √(b² - 4ac)) / (2a)

The quadratic formula is used primarily to solve quadratic equations, which are equations where the highest degree of the variable is 2. It is especially useful when the quadratic cannot be factored easily or when solutions are irrational or complex. By substituting the coefficients a, b, and c into the formula, one can determine the roots of the quadratic equation quickly and accurately.

For example, consider the quadratic equation 2x² - 4x - 6 = 0. To solve using the quadratic formula, identify the coefficients: a = 2, b = -4, and c = -6. Plugging into the formula:

x = [-(-4) ± √((-4)² - 4(2)(-6))] / (2 * 2)

x = (4 ± √(16 - (-48))) / 4

x = (4 ± √(16 + 48)) / 4

x = (4 ± √64) / 4

x = (4 ± 8) / 4

Calculating the two solutions:

x = (4 + 8) / 4 = 12 / 4 = 3.00

x = (4 - 8) / 4 = -4 / 4 = -1.00

Thus, the solutions rounded to the nearest hundredth are x = 3.00 and x = -1.00.

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It indicates the nature and number of solutions to the quadratic equation:

Discriminant, D = b² - 4ac

If D > 0, the quadratic has two real and distinct solutions. If D = 0, the quadratic has exactly one real solution, which is repeated (a repeated root). If D

References

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