3You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions.However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as You can use the imaginary unit to write the square root of any negative number.
10Solving a Quadratic Equation with Imaginary Solutions Solve the equation.This is just like solving x2 = You take theSquare root of both sides, making sure youconsider both the positive and negative roots.Checkx2 = –144–144(12i)2144i 2144(–1)x2 = –144–144144(–1)144i 2(–12i)2
11Example 2: Solve the equation. 5x2 + 90 = 0 5x2 + 90 = 0 Check What happens here?What happens?Solve by taking square roots.Simplify.5x = 05(18)i 2 +9090(–1) +90Check
15A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = The set of real numbers is a subset of the set of complex numbers C.Every complex number has a real part a and an imaginary part b.
16Real numbers are complex numbers where b = 0 Real numbers are complex numbers where b = 0. Imaginary numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pure imaginary numbers. For example, 7 can be written as 7 = 7 + 0i. (7 is the real part a, and 0i is the complex part b).Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
17In the complex number 3x – 5yi, what is the real part? What is the complex part?
18Example 3: Equating Two Complex Numbers Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true .Real parts4x + 10i = 2 – (4y)iImaginary partsSet the imaginary parts equal to each other.Set the real parts equal to each other.10 = –4y4x = 2Solve for y.Solve for x.
19Find the values of x and y that make each equation true. Example 3aFind the values of x and y that make each equation true.2x – 6i = –8 + (20y)iReal parts2x – 6i = –8 + (20y)iImaginary partsEquate the imaginary parts.Equate the real parts.2x = –8–6 = 20ySolve for y.Solve for x.x = –4
20Your turnFind the values of x and y that make each equation true.–8 + (6y)i = 5x – i
21Find the zeros of the function. f(x) = x2 + 10x + 26x2 + 10x + 26 = 0Set equal to 0.x2 + 10x = –26 +Rewrite.Add to both sides.x2 + 10x + 25 = –(x + 5)2 = –1Factor.Take square roots.Simplify.
22Find the zeros of the function. g(x) = x2 + 4x + 12Set equal to 0.x2 + 4x + 12 = 0x2 + 4x = –12 +Rewrite.Add to both sides.x2 + 4x + 4 = –12 + 4(x + 2)2 = –8Factor.Take square roots.Simplify.
23Find the zeros of the function. Example 4aFind the zeros of the function.f(x) = x2 + 4x + 13Set equal to 0.x2 + 4x + 13 = 0x2 + 4x = –13 +Rewrite.x2 + 4x + 4 = –13 + 4Complete the square.Factor.(x + 2)2 = –9Take square roots.x = –2 ± 3iSimplify.
24Your turnFind the zeros of the function.g(x) = x2 – 8x + 18
25The solutions and are related The solutions and are related. These solutions are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a – bi.If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates.When given one complex root, you can always find the other by finding its conjugate.Helpful Hint
26Example 5: Finding Complex Zeros of Quadratic Functions Find each complex conjugate.B. 6iA iWrite as a + bi.Write as a + bi.0 + 6i8 + 5i0 – 6iFind a – bi.8 – 5iFind a – bi.–6iSimplify.Reminder: ONLY the imaginary parts of complexconjugates are opposites.
27Your turnFind each complex conjugate.A. 9 – iB.C. –8i