Math  Help

The Quadratic Formula

What is the quadratic?

The quadratic formula is used to solve quadratic equations of the standard form, ax2 + bx + c = 0 where a, b and c are constant values with the unknown x.

The quadratic formula just like trinomial factoring and completing the square is used to determine the roots of a function. Recall, the roots are the points where the function touches the x-axis where the corresponding y value is equal to zero. Hence, for the quadratic formula to work, it is important that the function is set to zero.

For example: Given the quadratic equation:
ax2 + bx = - c.

First, set the equation equal to zero by bringing the c to the left side.

Equation for the quadratic formula

The quadratic formula is as follows:

where a is the coeffcient of the x2 b is the coeffcient of the x and c is the number without any variable beside it.

Example of using the quadratic formula

1. Determining two roots using the quadratic formula

Note the quadratic formula has the symbol ±. This means there are two possible values of x hence it is possible to have two roots of a quadratic function.

Example: Solve using the quadratic formula

5x2 + 6x + 1 = 0

Here: a = 5 b = 6 c = 1

Hence, the two roots of the quadratic function are x = and x = -1

2. Setting the equation to zero to use the quadratic formula

Example: Solve using the quadratic formula

x2 = 3x – 1

recall: to use the quadratic formula the quadratic equation must equal to zero.

Hence, we can move the 3x and -1 to the left side of the equal sign.

x2 – 3x + 1 = 0

Notice, there is no number in front of x2 so the coefficient must be 1. Also, we include the sign in front as part of the coefficient.

The b must be – 3 (not just 3).

a = 1 b = -3 c = 1

plugging the values into the quadratic formula:

The two roots of the quadratic function are: and

3. Turning the equation to standard form to use the quadratic formula

Example: f(x) = -2(x+6)2 + 4

The quadratic equation of the vertex form must be expanded first into standard form.

Finding the greatest common factor (GCF) results in smaller values for a, b and c hence.

a = 1 b = 12 c = 34

Therefore, the roots of the function: and

4. The quadratic function may have only one root

Example: Solve using the quadratic formula

2x2 + 4x + 4 = 0

In this scenario, the value under the square root is zero hence adding or subtracting the value makes no difierence. There is one root of the function 2x2 + 4x + 4 = 0 at x = -1.

5. The quadratic function may have zero root

Example: y = 3x2 + 4x + 2

a = 3 b = 4 c = 2

Since a square root of a negative number is undefined and does not result in a real number. Hence, in this case we are adding and subtracting a number that does not exist with the – 4. Therefore, the function has no real roots.

Take home points for using the quadratic formula

1. Make sure the given function is a quadratic equation Ex. Given y = x + 1. We cannot solve for x using the quadratic formula since it is a linear function

2. The function must be of the form: ax2 + bx + c = 0

3. Set the equation equal to zero before inputting values for a, b and c

4. Factor the “a” if possible, to work with smaller numbers

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