Which of the following is the formula for the solutions of a quadratic equation [tex]a x^2+ b x+c=0[/tex]?
A. [tex]\frac{b \pm \sqrt{b^2-4 a c}}{2 a}[/tex]
B. [tex]\frac{b \pm \sqrt{b^2+4 a c}}{2 a}[/tex]
C. [tex]\frac{-b \pm \sqrt{b^2+4 a c}}{2 a}[/tex]
D. [tex]\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}[/tex]
Answer (2)
Recall the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Compare the formula with the given options.
Identify the correct option.
The correct formula is 2 a − b ± b 2 − 4 a c .
Explanation
Understanding the Problem We are asked to identify the correct formula for the solutions of a quadratic equation of the form a x 2 + b x + c = 0 . The options given are variations of the quadratic formula.
Recalling the Quadratic Formula The quadratic formula is a well-known formula used to find the solutions (also called roots) of a quadratic equation. It is derived by completing the square on the general form of the quadratic equation. The formula is: x = 2 a − b ± b 2 − 4 a c
Comparing with the Options Now, we compare the correct quadratic formula with the given options:
Option 1: 2 a b ± b 2 − 4 a c - Incorrect (sign of b is wrong) Option 2: 2 a b ± b 2 + 4 a c - Incorrect (sign of b is wrong and sign inside the square root is wrong) Option 3: 2 a − b ± b 2 + 4 a c - Incorrect (sign inside the square root is wrong) Option 4: 2 a − b ± b 2 − 4 a c - Correct
Final Answer The correct formula for the solutions of a quadratic equation a x 2 + b x + c = 0 is: 2 a − b ± b 2 − 4 a c Therefore, the correct answer is the fourth option.
Examples
The quadratic formula is used in many fields, such as physics, engineering, and economics, to solve problems involving quadratic relationships. For example, in physics, it can be used to calculate the trajectory of a projectile, and in engineering, it can be used to design bridges and buildings. In economics, it can be used to model supply and demand curves. Understanding the quadratic formula allows us to solve real-world problems involving quadratic equations, making it a fundamental tool in various disciplines.
The correct formula for the solutions of a quadratic equation a x 2 + b x + c = 0 is given by 2 a − b ± b 2 − 4 a c . This corresponds to Option D in the provided choices. This formula allows us to compute the roots of the equation efficiently, applying the coefficients a, b, and c appropriately.
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