![]() ![]() ![]() Now, we can multiply through the parenthesis in ( 5 + □ ) ( 5 − □ ) and simplify: Treating ( 5 + □ ) and ( 5 − □ ) as single terms, we can multiply through the parenthesis to obtain Since the coefficient of □ is equal to 1, we can write the equation as ![]() Hence, its roots will be 5 + □ and 5 − □. We recall the conjugate root theorem, which states that the complex roots of a quadratic equation with real coefficients occur in complex conjugate pairs. In this example, we are given that a quadratic equation with real coefficients has a complex root. To reconstruct an equation given one of its complex roots as the next example will demonstrate.Įxample 5: Reconstructing a Quadratic Equation from a Complex Rootįind the quadratic equation □ + □ □ + □ = 0 with real coefficients □ and □ that has 5 + □ as one of its roots. We can also use our knowledge of the roots of quadratic equations with real coefficients We remember that complex numbers are equal to each other if their real parts are equal and their imaginary parts are equal. Furthermore, since a quadratic equation only has two roots, □ + □ □ must be the conjugate of □ + □ □. Since □ ≠ 0, we know that □ + □ □ is a complex root to a quadratic equation with real coefficients. What conditions, if any, must □, □, □, and □ satisfy? Answer The complex numbers □ + □ □ and □ + □ □, where □, □, □,Īre the roots of a quadratic equation with real coefficients. We can summarize the result of the previous example as follows.Įxample 4: Complex Roots of Quadratic Equations However, since □ is a root, we know that □ □ + □ □ + □ = 0 . We also know from the properties of the complex conjugate that for any two complex numbers □, and □ are real numbers, we can rewrite this as ![]() Given that any real number is equal to its complex conjugate and that □, įrom the properties of the complex conjugate, we know that for any two complex numbers Using the properties of the complex conjugate, we will provide a proof of this Let us examine whether or not these complex roots must be complex conjugates. If the discriminant is negative, then we know that the quadratic equation will have complex roots. Recall that the discriminant of a quadratic equation □ □ + □ □ + □ = 0 is the expression □ − 4 □ □ . If the discriminant of a quadratic equation with real coefficients is negative, will its complex roots be a conjugate pair? Answer In the next example, we will examine this fact in detail.Įxample 3: Conditions on the Roots of Quadratic Equations Inįact, it is true for any quadratic equation with real coefficients which has complex solutions. The fact that the roots of this equation are a complex conjugate pair is no accident. When we examine the solutions closely, we can notice that the two complex solutions are conjugates of each other. In the previous example, we observed that the quadratic equation had two complex solutions. Hence, we have two solutions for the quadratic equation: Recalling the property of complex numbers for a positive number □, Substituting these values into the quadratic formula, we have The given quadratic equation has the values □ = 1, □ = − 4, and □ = 8. Recall the quadratic formula for solving the quadratic equation □ □ + □ □ + □ = 0 : Solve the quadratic equation □ − 4 □ + 8 = 0 . Example 2: Solving Quadratic Equations with Complex Roots ![]()
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