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{{ option.icon }} {{ option.label }} # Using the Quadratic Formula to find Complex Roots

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### Direct messages

Sometimes, quadratic equations have no solutions that can be expressed using real numbers. However, it is possible to use complex numbers to express these non-real solutions.

### Rule

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
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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
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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
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The square can now be completed by adding the constant 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
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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
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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.

## Number of Solutions of a Quadratic Equation

The solutions of a quadratic equation can be interpreted graphically as the zeros of the related quadratic function. Therefore, the number of solutions of a quadratic equation is the same as the number of zeros of the related function.
If the function has two zeros, the equation ax2+bx+c=0 has two solutions. Similarly, if the function has one zero, the equation has one solution. Finally, if the function does not have any zeros, the equation has no real solutions.

## 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. ## Complex Solutions of a Quadratic Equation

When the discriminant is negative, there are no real solutions of the quadratic equation. However, the square root of a negative number can be written as an imaginary number. This way, complex solutions of a quadratic equation can be found.

## Find the complex solutions of the quadratic equation

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Solve the equation using the quadratic formula.
2x24x+10=0
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Since the quadratic equation is given in standard form, we can immediately identify the constants
Let's substitute these values into the quadratic formula to solve the equation.
2x24x+10=0
Notice that there is a negative number inside the square root. Thus, the equation has no real solutions. However, we can calculate the complex roots using the identity
x=1±2i
The solutions to the equation are x=1+2i and x=12i.