Quadratic equations are mathematical equations of the form ax^2 + bx + c = 0, where x is the variable, and a, b, and c are constants. One of the most useful tools in solving quadratic equations is the quadratic formula.

The quadratic formula is given by:

x = (-b ± sqrt(b^2 – 4ac)) / 2a

This formula gives the solutions to any quadratic equation of the form ax^2 + bx + c = 0. The solutions to the quadratic equation are given by the values of x that satisfy the equation.

How to Use the Quadratic Formula

To use the quadratic formula, we need to follow a few steps:

• Step 1: Identify the values of a, b, and c in the quadratic equation.
• Step 2: Substitute the values of a, b, and c into the quadratic formula.
• Step 3: Simplify the expression inside the square root and evaluate it.
• Step 4: Solve for x by performing the operations indicated in the formula.

Let’s look at an example to see how the quadratic formula is used to solve quadratic equations.

Example: Solve the quadratic equation 2x^2 – 3x – 5 = 0 using the quadratic formula.

Step 1: Identify the values of a, b, and c in the quadratic equation.

In this case, a = 2, b = -3, and c = -5.

Step 2: Substitute the values of a, b, and c into the quadratic formula.

We get:

x = (-(-3) ± sqrt((-3)^2 – 4(2)(-5))) / 2(2)

Simplifying this expression gives us:

x = (3 ± sqrt(9 + 40)) / 4

x = (3 ± sqrt(49)) / 4

Step 3: Simplify the expression inside the square root and evaluate it.

The expression inside the square root simplifies to 49, so we have:

x = (3 ± 7) / 4

Step 4: Solve for x by performing the operations indicated in the formula.

We get two solutions:

x = (3 + 7) / 4 = 5/2

x = (3 – 7) / 4 = -1

Therefore, the solutions to the quadratic equation 2x^2 – 3x – 5 = 0 are x = 5/2 and x = -1.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It is important to follow the steps outlined above to use the formula correctly. By identifying the values of a, b, and c in the quadratic equation, substituting them into the quadratic formula, simplifying the expression inside the square root, and solving for x, we can find the solutions to any quadratic equation.

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