Quadratic Equations

Algebra I

Vocabulary

• Quadratic Function (equation) – A function describing the height of a rocket.

•Standard Form y = ax²+ bx + c, a 0

•Parabola – The graph of a quadratic function.

Vocabulary

• Solutions – Called roots, zeros or x intercepts. The point(s) where the parabola crosses the x axis.

• Minimum – The lowest point of the parabola on the y axis (on the calculator, when the y value is the lowest).

• Maximum – The highest point of the parabola on the y axis (on the calculator, when the y value is the highest).

Vocabulary

• Number of roots – Can have one, two or no roots.

• Vertex – The minimum or maximum point.

• Positive Parabola – Going up.

• Negative Parabola – Going down.

Vocabulary

• Axis of Symmetry – Directly down the middle of the parabola. The only point in the middle of the parabola is the vertex. Each point on the parabola that is on one side of the axis of symmetry has a corresponding point on the other side of the axis of symmetry.

Axis of Symmetry

• Axis of symmetry equation – finding the x value.

x = Ex) y = -3x² – 6x + 4a = -3b = -6c = 4

Axis of Symmetry

• Axis of symmetry equation – finding the x value.

Ex) y = -3x² – 6x + 4a = -3 x = -(-6)/2(-3)b = -6 x = -1c = 4

Axis of Symmetry

• Find the y value by substituting the x value into the equation and solve for y.

Ex) y = -3x² – 6x + 4 x = -1 y = -3(-1)²- 6(-1) + 4 y = 7 Vertex (-1, 7)

Axis of Symmetry

• Calculator– Enter the equation into the y= function on

the calculator– Look at the graph to determine a positive or

negative parabola– Go to 2nd graph to see the table

• Scroll up and down to find where the y values start to repeat, there will be one point that doesn’t repeat, this is the vertex.

• The number in the x column is the x value and the corresponding number in the y column is the y value.

Example

y = x²- x - 6

Example

y = x²- x – 6 (not found in calculator)a = 1 x = -(-1)/2(1)b = -1 x = ½ c = -6 y = (½)²- ½ - 6 y = -6¼

Vertex (½, -6¼)

Now you try…

y = 2x²- 4x – 5

y = -x²+ 4x - 1

Now you try…

y = 2x²- 4x – 5 (1,-7) minimum

y = -x²+ 4x – 1

(2,3) maximum

Roots

• Can solve by factoring x²+ 6x – 7 = 0

Roots

• Can solve by factoring x²+ 6x – 7 = 0 (x + 7)(x – 1) = 0

Roots

• Can solve by factoring x²+ 6x – 7 = 0 (x + 7)(x – 1) = 0

Now set each factor =0

Roots

• Can solve by factoring x²+ 6x – 7 = 0 (x + 7)(x – 1) = 0

x + 7 = 0 x – 1 = 0 x = -7 x = 1*** Two roots (sometimes called a double root)

Roots

b²+ 4b = -4 (re write =0) b²+ 4b + 4 = 0

Roots

b²+ 4b = -4 (re write =0) b²+ 4b + 4 = 0 Now factor

Roots

b²+ 4b = -4 (re write =0) b²+ 4b + 4 = 0 Now factor(b + 2)(b + 2) = 0 b + 2 = 0 b = -2***This is a single root, only one answer.

Roots

• Roots can often times be found on the calculator– Enter the equation in the y=– Graph, this will show you how many

roots– Go to the table– Find where the y value is zero

Roots

x²- x + 4 = 0

Roots

x²- x + 4 = 0 Prime, can’t factor

No x intercept – no roots (see this when you graph) – always check the graph, some are prime, but still cross the x axis.

Roots

n²+ 6n + 7 = 0

• Prime – can’t factor, but it does cross the x axis.

• Sometimes we estimate the roots.• One root is between -5 and -4, the

other root is between -2 and -1.• These are rational roots.

Roots

• When you can’t factor the equation, use the quadratic formula:

*** This is on your formula sheet for testing.

Roots

• Use the Quadratic Formula:24x²- 14x = 6 (re write)24x²- 14x – 6 = 0a = 24, b = -14, c = -6

*** This is on your formula sheet for testing.

Roots

• Use the Quadratic Formula:24x²- 14x – 6 = 0a = 24, b = -14, c = -6

Solve

Roots

• Use the Quadratic Formula:24x²- 14x – 6 = 0

Roots

• Use the Quadratic Formula:24x²- 14x – 6 = 0

Now separate into two problems. One is +, the other is -

Roots

• Use the Quadratic Formula:24x²- 14x – 6 = 0

Solve both problems

Roots

• Use the Quadratic Formula:24x²- 14x – 6 = 0

x ≈ -0.3 x ≈ 0.9

**** The symbol ≈ means approximate value.

Now you try…

x²- 2x – 24 = 0

3x²+ 5x + 11 = 0

Now you try…

x²- 2x – 24 = 0 {-4,6} This is a solution set, NOT an ordered pair. These are the numbers where the parabola crosses the x axis. 3x²+ 5x + 11 = 0 No roots – no solution

Discriminant

• To find the discriminant use part of the quadratic formula.

b²- 4ac

Discriminant

• If the discriminant is < 0, there are no roots.

• If the discriminant is > 0, there are two roots.

• If the discriminant is = 0, there is one root.