Presentation on theme: "Taking the Square Root of Both Sides Adapted from Walch EducationAdapted from Walch Education."— Presentation transcript:
Taking the Square Root of Both Sides Adapted from Walch EducationAdapted from Walch Education
Complex Numbers The imaginary unit i represents the non-real value. i is the number whose square is –1. We define i so that and i 2 = –1. A complex number is a number with a real component and an imaginary component. Complex numbers can be written in the form a + bi, where a and b are real numbers, and i is the imaginary unit : Taking the Square Root of Both Sides2
Real Numbers Real numbers are the set of all rational and irrational numbers. Real numbers do not contain an imaginary component. Real numbers are rational numbers when they can be written as, where both m and n are integers and n ≠ 0. Rational numbers can also be written as a decimal that ends or repeats : Taking the Square Root of Both Sides3
Irrational Numbers Real numbers are irrational when they cannot be written as, where m and n are integers and n ≠ 0. Irrational numbers cannot be written as a decimal that ends or repeats. The real number is an irrational number because it cannot be written as the ratio of two integers. examples of irrational numbers include and π : Taking the Square Root of Both Sides4
Quadratic Equations A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0, where a ≠ 0. Quadratic equations can have no real solutions, one real solution, or two real solutions. When a quadratic has no real solutions, it has two complex solutions. Quadratic equations that contain only a squared term and a constant can be solved by taking the square root of both sides. These equations can be written in the form x 2 = c, where c is a constant : Taking the Square Root of Both Sides5
Quadratic Equations c tells us the number and type of solutions for the equation : Taking the Square Root of Both Sides6 c Number and type of solutions NegativeTwo complex solutions 0 One real, rational solution Positive and a perfect square Two real, rational solutions Positive and not a perfect square Two real, irrational solutions
Practice # 1 Solve (x – 1) = –1 for x : Taking the Square Root of Both Sides7
Solution Isolate the squared binomial. Use a square root to isolate the binomial : Taking the Square Root of Both Sides8 (x – 1) = –1Original equation (x – 1) 2 = –16 Subtract 15 from both sides.
Solution, continued Simplify the square root. There is a negative number under the radical, so the answer will be a complex number : Taking the Square Root of Both Sides9 Equation Write –16 as a product of a perfect square and –1. Product Property of Square Roots Simplify.
Solution, continued Isolate x. The equation (x – 1) = –1 has two solutions, 1 ± 4i : Taking the Square Root of Both Sides10 Equation Add 1 to both sides.
Try This One… Solve 4(x + 3) 2 – 10 = –6 for x : Taking the Square Root of Both Sides11