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Quadratic Functions and Equations

Using the Quadratic Formula to find Complex Roots

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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

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
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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.

Concept

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.

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

Method

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.

Example

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.
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