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We will use the example to illustrate each step.
Given a quadratic equation that cannot be factored, and with first add or subtract the constant term to the right sign of the equal sign.
Multiply the b term by and square it.
Add to both sides of the equal sign and simplify the right side. We have
The left side of the equation can now be factored as a perfect square.
Use the square root property and solve.
The solutions are
Solve the quadratic equation by completing the square:
First, move the constant term to the right side of the equal sign.
Then, take of the b term and square it.
Add the result to both sides of the equal sign.
Factor the left side as a perfect square and simplify the right side.
Use the square root property and solve.
The solutions are
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,
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
Solve the quadratic equation:
Identify the coefficients: Then use the quadratic formula.
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