The fourth method of solving a
quadratic equation is by using the
quadratic formula , a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.
We can derive the quadratic formula by
completing the square . We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by
and obtain a positive
a . Given
we will complete the square as follows:
First, move the constant term to the right side of the equal sign:
As we want the leading coefficient to equal 1, divide through by
a :
Then, find
of the middle term, and add
to both sides of the equal sign:
Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:
Now, use the square root property, which gives
Finally, add
to both sides of the equation and combine the terms on the right side. Thus,
The quadratic formula
Written in standard form,
any quadratic equation can be solved using the
quadratic formula :
where
a ,
b , and
c are real numbers and
Given a quadratic equation, solve it using the quadratic formula
Make sure the equation is in standard form:
Make note of the values of the coefficients and constant term,
and
Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.
Calculate and solve.
Solve the quadratic equation using the quadratic formula
Solve the quadratic equation:
Identify the coefficients:
Then use the quadratic formula.