9.1 Square Roots and the Pythagorean Theorem

9.1 Square Roots and the Pythagorean Theorem By definition Ö25 is the number you would multiply times itself to get 25 for an answer. Because we are familiar with multiplication, we know that Ö25 = 5 Numbers like 25, which have whole numbers for their square roots, are called perfect squares You need to memorize at least the first 15 perfect squares

Square root Square root Perfect square Perfect square 1 81 Ö1 = 1 Ö81 = 9 4 100 Ö4 = 2 Ö100 = 10 9 121 Ö9 = 3 Ö121 = 11 16 144 Ö16 = 4 Ö144 = 12 25 169 Ö25 = 5 Ö169 = 13 36 196 Ö36 = 6 Ö196 = 14 49 225 Ö49 = 7 Ö225 = 15 64 Ö64 = 8

9.1 Square Roots and the Pythagorean Theorem Every whole number has a square root Most numbers are not perfect squares, and so their square roots are not whole numbers. Most numbers that are not perfect squares have square roots that are irrational numbers Irrational numbers can be represented by decimals that do not terminate and do not repeat The decimal approximations of whole numbers can be determined using a calculator

9.1 Square Roots and the Pythagorean Theorem Find the square roots of the given numbers If the number is not a perfect square, use a calculator to find the answer correct to the nearest hundredth. 81 37 158

9.1 Square Roots and the Pythagorean Theorem Find the square roots of the given numbers If the number is not a perfect square, use a calculator to find the answer correct to the nearest thousandth.

c a b 9.1 Square Roots and the Pythagorean Theorem The Pythagorean Theorem For any right triangle, the sum of the squares of the lengths of the legs a and b, equals the square of the length of the hypotenuse. a2 + b2 = c2

c 6 8 9.1 Square Roots and the Pythagorean Theorem Find c. a2 + b2 = c2

c 4 6 9.1 Square Roots and the Pythagorean Theorem Find c. a2 + b2 = c2

17 a 8 9.1 Square Roots and the Pythagorean Theorem Find a. a2 + b2 = c2

60 ft 60 ft 9.1 Square Roots and the Pythagorean Theorem The length of each side of a softball field is 60 feet. How far is it from home to second? 602 + 602 = c2 3600 + 3600 = 7200 c2 = 7200

9.2 Solving Quadratic Equations Solving x2 = d by Finding Square Roots • If d is positive, then x2 = d has two solutions • The equation x2 = 0 has one solutions: • If d is negative, then x2 = d has no solution.

9.2 Solving Quadratic Equations Solve • x2 = 49 • x2 = 12 • x2 = 0 • x2 = -9

9.2 Solving Quadratic Equations Solve • 3x2 +1 = 76 • 4x2 + 6 = 70 • 5x2 – 7 = 18 • 3x2 – 10 = 38

9.3 Graphing Quadratic Equations y = ax2 + bx + c

y Vertex x Vertex 9.3 Graphing Quadratic Equations The graph of a quadratic function is a parabola. A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex. If the parabola opens down, the vertex is the highest point. NOTE: if the parabola opened left or right it would not be a function!

y a > 0 x a < 0 9.3 Graphing Quadratic Equations The parabola will open up when the a value is positive. The standard form of a quadratic function is y = ax2 + bx + c The parabola will open down when the a value is negative.

Line of Symmetry y x 9.3 Graphing Quadratic Equations Parabolas have a symmetric property to them. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the line of symmetry. Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side. The line of symmetry ALWAYS passes through the vertex.

9.3 Graphing Quadratic Equations When a quadratic function is in standard form For example… Find the line of symmetry of y = 3x2 – 18x + 7 y = ax2 + bx + c, The equation of the line of symmetry is Using the formula… This is best read as … the opposite of b divided by the quantity of 2 times a. Thus, the line of symmetry is x = 3.

9.3 Graphing Quadratic Equations y = –2x2 + 8x –3 We know the line of symmetry always goes through the vertex. STEP 1: Find the line of symmetry Thus, the line of symmetry gives us the x – coordinate of the vertex. STEP 2: Plug the x – value into the original equation to find the y value. y = –2(4)+ 8(2) –3 y = –2(2)2 + 8(2) –3 To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. y = –8+ 16 –3 y = 5 Therefore, the vertex is (2 , 5)

9.3 Graphing Quadratic Equations The standard form of a quadratic function is given by y = ax2 + bx + c STEP 1: Find the line of symmetry STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

y x 9.3 Graphing Quadratic Equations Let's Graph ONE! Try … y = 2x2 – 4x – 1 STEP 1: Find the line of symmetry Thus the line of symmetry is x = 1

y x 9.3 Graphing Quadratic Equations Let's Graph ONE! Try … y = 2x2 – 4x – 1 STEP 2: Find the vertex Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex. Thus the vertex is (1 ,–3).

y x 9.3 Graphing Quadratic Equations Let's Graph ONE! Try … y = 2x2 – 4x – 1 STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

9.3 Graphs of Quadratic Equations For y = -x2 - 2x + 8 identify each term, graph the equation, find the vertex, and find the solutions of the equation. -Vertex: x =(-b/2a) x= -(-2/2(-1)) x= 2/(-2) x= -1 Solve for y: y = -x2 -2x + 8 y = -(-1)2 -(2)(-1) + 8 y = -(1) + 2 + 8 y = 9 Vertex is (-1, 9)

9.4 The Quadratic Formula The quadratic formula allows you to find the roots of a quadratic equation (if they exist) even if the quadratic equation does not factorise. The formula states that for a quadratic equation of the form : ax2 + bx + c = 0 The roots of the quadratic equation are given by :

The Quadratic Formula.

Example 1 Use the quadratic formula to solve the equation : x2 + 5x + 6 = 0 Solution: x2 + 5x + 6= 0 a = 1 b = 5 c = 6 x = - 2 or x = - 3 These are the roots of the equation.

Example 3 Use the quadratic formula to solve the equation : 8x2 - 22x + 15= 0 Solution: 8x2 - 22x + 15= 0 a = 8 b = -22 c = 15 x = 3/2 or x = 5/4 These are the roots of the equation.

Example 4 Use the quadratic formula to solve for x to 2 decimal places. 2x2 + 3x - 7= 0 Solution: 2x2 + 3x – 7 = 0 a = 2 b = 3 c = - 7 x = 1.27 or x = - 2.77 These are the roots of the equation.

Discriminant - ± - 2 b b 4 ac = x 2 a 9.5 Problem Solving Using the Discriminant

9.5 Problem Solving Using the Discriminant The number of solutions in a quadratic equation Consider the equation ax2 + bx + c = 0 • If b2 – 4ac > 0, then the equation has 2 solutions. • If b2 – 4ac = 0, then the equation has 1 solution. • If b2 – 4ac < 0, then the equation has no solution.

9.5 Problem Solving Using the Discriminant Find the discriminant of 3x2 + x – 2 = 0 and tell the nature of its roots. Discriminant = b2 – 4ac = 12 – 4(3)(-2) = 1 – (-24) = 1 + 24 = 25 So, there are two solutions

9.5 Problem Solving Using the Discriminant Determine the number of solutions • 2x2 – x + 3 = 0 • x2 + x + 4 = 0 • 3x2 –5x - 3 = 0 • 2x2 – x - 9 = 0

9.5 Problem Solving Using the Discriminant Match the discriminant with the graph • b2 – 4ac = 7 b. b2 – 4ac = -2 c. b2 – 4ac = 0

9.6 Graphing Quadratic Inequalities • Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is < or >. Use a solid line if the inequality is ≤ or ≥. • Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point. Steps for Drawing the Graph of an Inequality in Two Variables

9.6 Graphing Quadratic Inequalities Solve y > x2

9.1 Square Roots and the Pythagorean Theorem