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# Quadratic Equations with Imaginary Solutions

Recall that a quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a 0.

Any quadratic equation can be solved using the quadratic formula:

You probably know that if the discriminant, b2 - 4ac, is negative then the equation has no real number solutions.

Now we know that the square root of a negative number is an imaginary number. In particular, if the discriminant b2 - 4ac < 0 then the quadratic equation has two imaginary solutions. In fact, the solutions are complex conjugates.

Example 1

Solve using the quadratic formula: x2 + 2x = -5

Solution

 Step 1 Write the quadratic equation in standard form. Add 5 to both sides of the equation. x2 + 2x = -5 Step 2 Identify the values of a, b, and c. a = 1, b = 2, c = 5 Step 3 Substitute the values of a, b, and c into the quadratic formula. Substitute 1 for a, 2 for b, and 5 for c. Step 4 Simplify. Simplify the radicand and the denominator. Use to simplify the square root. Cancel the common factor, 2, in the numerator and denominator. x = -1 Â± 2i Step 5 Check each solution. We leave the check to you.
So, the solutions of x2 + 2x = -5 are -1 + 2i and -1 - 2i.

Notice that -1 +2i and -1 - 2i are complex conjugates.

Note:

• x2 + 2x + 5 = 0 The discriminant of the equation is b2 - 4ac = 22 - 4(1)(5) = -16. Since the discriminant is negative, we know that the two solutions will be complex conjugates.
• Be careful when you cancel. Be sure to divide each term of the numerator by 2.
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