Introduction You can determine how far a ladder will extend from the base of a wall by creating a quadratic equation and then taking the square root. To find this length, you only need to find the positive square root because a negative distance would not make sense in this situation. For certain types of quadratics, we can solve by taking the square root, but in most problems, we need to take both the positive and negative square root : Taking the Square Root of Both Sides

Key Concepts 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. An imaginary number is any number of the form bi, where b is a real number,, and b ≠ 0. 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. For example, 5 + 3i is a complex number. 5 is the real component and 3i is the imaginary component : Taking the Square Root of Both Sides

Key Concepts, continued 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. The real number 0.4 is a rational number because it can be written as the fraction : Taking the Square Root of Both Sides

Key Concepts, continued 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. Other examples of irrational numbers include and π : Taking the Square Root of Both Sides

Key Concepts, continued 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 Sides

Key Concepts, continued When we take the square root of both sides, we need to remember that a number and its opposite have the same square. Therefore, rather than simply taking the positive square root, we need to take the positive and negative square root. For x 2 = c, we find that We can use a similar method to solve quadratic equations in the form (ax + b) 2 = c. c tells us the number and type of solutions for the equation : Taking the Square Root of Both Sides

Key Concepts, continued : Taking the Square Root of Both Sides 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

Common Errors/Misconceptions forgetting to use ± and therefore forgetting that there may be two solutions taking the square root before isolating the squared term : Taking the Square Root of Both Sides

Guided Practice Example 2 Solve (x – 1) = –1 for x : Taking the Square Root of Both Sides

Guided Practice: Example 2, continued 1.Isolate the squared binomial : Taking the Square Root of Both Sides (x – 1) = –1Original equation (x – 1) 2 = –16 Subtract 15 from both sides.

Guided Practice: Example 2, continued 2.Use a square root to isolate the binomial. Take the square root of both sides. Remember to use the ± sign : Taking the Square Root of Both Sides

Guided Practice: Example 2, continued 3.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 Sides Equation Write –16 as a product of a perfect square and –1. Product Property of Square Roots Simplify.

Guided Practice: Example 2, continued 4.Isolate x. The equation (x – 1) = –1 has two solutions, 1 ± 4i : Taking the Square Root of Both Sides Equation Add 1 to both sides. ✔

Guided Practice: Example 2, continued : Taking the Square Root of Both Sides

Guided Practice Example 3 Solve 4(x + 3) 2 – 10 = –6 for x : Taking the Square Root of Both Sides

Guided Practice: Example 3, continued 1.Isolate the squared binomial : Taking the Square Root of Both Sides 4(x + 3) 2 – 10 = –6Original equation 4(x + 3) 2 = 4Add 10 to both sides. (x + 3) 2 = 1Divide both sides by 4. Take the square root of both sides.

Guided Practice: Example 3, continued 2.Isolate x : Taking the Square Root of Both Sides Equation Subtract 3 from both sides.

Guided Practice: Example 3, continued 3.Split the answer into two separate expressions and evaluate. x = –3 + 1 = –2 x = –3 – 1 = –4 The equation 4(x + 3) 2 – 10 = –6 has two solutions, –2 and – : Taking the Square Root of Both Sides ✔

Guided Practice: Example 3, continued : Taking the Square Root of Both Sides