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Quadratic Equations

Solving Quadratic Equations using the Quadratic Formula

The general form of a quadratic equation is
ax2+bx+c=0,
where a, b, and c are real numbers, and a0. There are many ways to solve such equations. One way is by using the quadratic formula, which is derived by completing the square.

Rule

Quadratic Formula

To solve a quadratic equation written in standard form ax2+bx+c=0, the Quadratic Formula can be used.

In this formula, the discriminant b24ac determines the number of real solutions of the quadratic equation.

Proof

The Quadratic Formula can be derived by completing the square given the standard form of the quadratic equation ax2+bx+c=0. This method will be used to isolate the x-variable. To complete the square, there are five steps to follow.

1

Factor Out the Coefficient of x2
It is easier to complete the square when the quadratic expression is written in the form x2+bx+c. Therefore, the coefficient a should be factored out.
Since the equation is quadratic, the coefficient a is not equal to 0. Therefore, both sides of the equation can be divided by a.

2

Identify the Constant Needed to Complete the Square
The next step is to rewrite the equation by moving the existing constant to the right-hand side. To do so, will be subtracted from both sides of the equation.
The constant needed to complete the square can now be identified by focusing on the x-term, while ignoring the rest. One way to find this constant is by squaring half the coefficient of the x-term, which in this case is
Note that leaving the constant as a power makes the next steps easier to perform.

3

Complete the Square
The square can now be completed by adding the found in Step 2 to both sides of the equation.
The first three terms form a perfect square trinomial, which can be factored as the square of a binomial. The other two terms do not contain the variable x. Therefore, their value is constant.

4

Factor the Perfect Square Trinomial
The perfect square trinomial can now be factored and rewritten as the square of a binomial.
The process of completing the square is now finished.

5

Simplify the Equation
Finally, the right-hand side of the equation can be simplified.
Simplify right-hand side
Now, there is only one x-term. To isolate x, it is necessary to take square roots on both sides of the equation. This results in both a positive and a negative term on the right-hand side.
Now, the equation can be further simplified to isolate x.
Solve for x
Finally, the Quadratic Formula has been obtained.

fullscreen
Exercise
Use the quadratic formula to solve the equation.
2x24x16=0
Show Solution
Solution
Notice that the given equation is written in standard form. Thus, it can be solved using the quadratic formula. To begin, it's necessary to note the values of a,b, and c. It can be seen that
To solve the equation, we can substitute these values into the formula and simplify. Remember, since a quadratic function can have 0,1, or 2 roots, this equation can have 0,1, or 2 real solutions.
2x24x16=0
The solutions to the equation are x=-2 and x=4.

Concept

Discriminant

In the Quadratic Formula, the expression b24ac, which is under the radical symbol, is called the discriminant.

A quadratic equation can have two, one, or no real solutions. Since the discriminant is under the radical symbol, its value determines the number of real solutions of a quadratic equation.

Value of the Discriminant Number of Real Solutions
b24ac>0 2
b24ac=0 1
b24ac<0 0

Moreover, the discriminant determines the number of x-intercepts of the graph of the related quadratic function.

The number of $x$-intercepts of the graph of a quadratic function
fullscreen
Exercise
Determine the number of real solutions to the equations without solving them.
Show Solution
Solution

It's possible to use the discriminant of the quadratic formula to determine the number of solutions a quadratic equation has. We'll first focus on x22x+9=0. Since the equation is written in standard form, we can see that a=1,b=-2, and c=9. We can substitute these values into the discriminant and simplify.

b24ac
(-2)2419
436
-32
Since -32<0, the first quadratic equation has 0 real solutions. We can perform the same process on the second equation where a=1,b=-4, and c=4.
b24ac
(-4)2414
1616
0
Since 0=0, the second quadratic equation has 1 real solution.
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