4.1 – Graphical Solutions of Quadratic Equations Date:

Key Ideas:

Quadratic functions are written as 𝑓(𝑥) = 𝑥2 − 𝑥 − 6 OR 𝑦 = 𝑥2 − 𝑥 − 6. f(x) is ‘f of x’ and means that the y value is dependent upon the value of x. Once you have an x value and you substitute it into the equation, the value of f(x) will result. f(x) is really just a another way of writing y, and is used when the graph is a function (passes the vertical line test).

A very important feature of the parabola that results from a quadratic function is where it touches or crosses the x-axis. These are the x-intercepts of the parabola. What is the y value at an x-intercept?

Therefore, to find the x-intercepts of a parabola, we can set y = 0 (or f(x) = 0) and solve the resulting equation.

When y is set to 0, we call the question a quadratic equation instead of a quadratic function.

Quadratic function: f(x) = x2 – x – 6 OR y = x2 – x – 6

Quadratic equation: x2 – x – 6 = 0

The x-intercepts of the parabola are the zeros of the quadratic function. They are also called the solutions or roots of the quadratic equation.

How many x-intercepts can a parabola have? Draw the possibilities:

Therefore, how many zeros can there be for a quadratic function?

How many roots or solutions can there be for a quadratic equation?

One method we can use to find the zeros of a quadratic function is to complete the square and turn standard form into vertex form, and then graph. Another way to graph a function in standard form is by using a table of values.

Example – What are the roots of the equation 𝑥2 − 𝑥 − 6 = 0 ?

1) Change the quadratic equation to a quadratic function. 2) Make a table of values and graph the function. 3) Note any x-intercepts. These are the zeros of the function and the roots of the

equation. 4) Check the solution(s).

Example – What are the roots of the equation −𝑥2 + 6𝑥 = 9 ?

Follow the same steps as above, except this time change to vertex form to graph, rather

than creating a table of values.

finding

roots by

graphing

Example - Solve 022 2 xx by graphing.

To ‘solve’ a quadratic equation means to find the roots, if any. In this example, graph the corresponding quadratic function using a table of values.

4.2A – Factoring & Solving Quadratic Equations Date:

Key Ideas:

You can often find the roots of a quadratic equation by factoring when in standard form

02 cbxax . Remember, the roots or solutions of the quadratic equation correspond to the zeros of the quadratic function, and the x-intercepts of the parabola.

if a = 1 Example - Solve and check 232 xx

To ‘solve’ a quadratic equation means to find the roots. Steps are as follows:

1) Get everything to one side so that only zero is on the other. 2) Identify a, b, and c values. 3) If a = 1, find two numbers that multiply to equal the c value that also add to equal the

b value. 4) Factor the trinomial into two binomials using the two numbers from step 3. 5) The roots are the x-values that will make the product of the two binomials zero. If

either of the binomials equal zero, then the product of the binomials will equal zero. Therefore, identify the x values that make each binomial equal to zero.

Check: Sketch the Graph:

Examples: a) Solve 84082 xx b) Solve 2𝑥2 + 6𝑥 − 108 = 0

if a ≠ 1 When 1a , the factoring method is called “decomposition.”

Example – Solve 0253 2 xx

1) Get all terms to one side so only zero is on the other. 2) Find the a, b, and c values. 3) Find the two numbers that multiply to equal ac and add to equal the b value. 4) Rewrite the quadratic equation but split the b term into the two numbers from step 3. 5) Now the equation has four terms so we can factor by grouping. Factor what you can

out of the first two terms, and then the second two. The resulting brackets must match.

6) Factor the identical brackets out, leaving the components of the second bracket. 7) Get the roots by finding what x value will make each bracket equal zero.

Examples - a) Solve 1532 2 xx b) Solve xx 7)2(3 2

If c = 0 Example - Solve xx 53 2

Trick for finding roots:

Example - Solve 0252 x .

1) If the quadratic equation does not have a middle term (bx term), check to see if the two terms are perfect squares. If they are, and there is a subtraction sign between them, you can factor using the difference of squares technique.

2) Make two binomial brackets. Put a ‘+’ sign in the middle of one, and a ‘-‘ sign in the middle of the other.

3) Square root the first term in the quadratic equation and put the square root in the first position of each binomial bracket.

4) Square root the second term in the quadratic equation and put the square root in the second position of each binomial bracket.

5) You can check for correctness by foiling to see if you end up with your original quadratic equation.

6) Find the roots for the quadratic equation.

Examples: a) Solve 9𝑝2 − 144 = 0 b) Solve 49 − 4𝑥2 = 0

Examples - a) Write a quadratic equation with roots 6 and -1. b) 4 and -4

difference

of squares

4.2B – Factoring & Solving Quadratic Equations Date:

Key Ideas:

Warmup 1 – Write a quadratic equation in standard form with the roots −4

5 and

1

2 .

Example – Solve 1

2𝑥2 − 𝑥 − 4 = 0

When you have an a value that’s a fraction, factor it out of the three terms, and then look

to factor the remaining trinomial in the brackets.

*If you divide by 1

2 , it is the equivalent of multiplying by…

Example – Solve 1

4𝑥2 − 𝑥 = 3

If a is a

fraction

Example – Solve −2(𝑥 + 3)2 + 12(𝑥 + 3) + 14 = 0

Notice the quadratic pattern. Substitute a variable in place of the brackets, factor, & solve the equation, and then substitute the brackets back in for the variable and solve for x.

Example – Solve 4(5𝑝 − 3)2 + 10(5𝑝 − 3) − 6 = 0

Example – Difference of Squares: Solve 9(2𝑡 + 1)2 − 4(𝑡 − 2)2 = 0

Example – A waterslide ends with the slider dropping into a deep pool. The path of the slider after leaving the slide can be approximated by the quadratic function

ℎ(𝑑) = −1

6𝑑2 −

1

6𝑑 + 2, where h is the height above the surface of the pool and d is the

horizontal distance the slider travels after leaving the slide, both in feet. What is the horizontal distance the slider travels before hitting the water?

Example – The area of a rectangular Ping-Pong table is 45𝑓𝑡2. The length is 4ft more than

the width. What are the dimensions of the table?

word problems

4.3A – Solving Quadratic Equations by Completing the Square Date:

Key Ideas:

When there is no bx term in a quadratic equation, first look to see if it is a difference of squares (is each term a perfect square, and is there a subtraction sign in between?). If it is not a difference of squares, it can be solved by the square root method.

The general form of a quadratic equation that you could solve using the square root method is 𝑎𝑥2 ± 𝑐 = 0.

Example – Solve 0219 2 y

1) Get everything that isn’t squared to one side. 2) Square root both sides. 3) Consider both the positive and negative square roots. 4) Simplify any radical solutions as much as possible.

Examples: a) 2𝑥2 − 11 = 87 b) 50𝑦2 = 72 c) 16)3( 2 x

square root method

Sometimes factoring quadratic equations (4.2) is not possible, as you cannot find the two numbers that multiply to c (or ac) and add to b. When this is the case, you can still solve the quadratic equation by a method called completing the square.

Example – Solve 𝑥2 − 24 = −10𝑥 by completing the square. This example can easily be solved by factoring, but we will use it to introduce how to complete the square.

1) Get c to one side of the equation. 2) If the a value is 1, find the b value, halve it, and square it. 3) Add the number from step 2 to BOTH sides of the equation. 4) Factor the trinomial on the left (we created a perfect square trinomial, so it will lead to 2

brackets that are exactly the same). 5) Solve using the square root principle.

Examples – Solve by completing the square. Express the solutions to b) in exact form.

a) 𝑤2 − 4𝑤 − 11 = 0 b) 𝑥2 + 5𝑥 + 7 = 0

completing the square when a = 1

4.3B – Solving Quadratic Equations by Completing the Square Part 2 Date:

Key Ideas:

If 𝑎 ≠ 1, there are a few more considerations when completing the square.

Example – Solve 0152 2 xx by completing the square

1) Get c to one side of the equation. 2) Factor the a value out of the left side.

3) Divide both sides by the a value to leave bxx 2 on the left side of the equation. THEN find the b value, halve it, and square it.

4) Add the number from step 3 to BOTH sides of the equation. 5) Factor the resulting trinomial on the left. 6) Solve using the square root principle.

Examples – Solve by completing the square. Answer b to the nearest hundredth.

a) 3𝑥2 − 2 = −4𝑥 b) −2𝑥2 − 3𝑥 + 7 = 0

completing the square when 𝒂 ≠ 𝟏

An easy way to halve a fraction is to double the denom-inator.

Example – Solve by completing the square: 3𝑥2 + 6𝑥 − 1 = 0

Butchart Gardens wants to build a pathway around its rose garden. The rose garden is currently 30m x 20m. The pathway will be built by extending each side by an equal amount. If the area of the garden and path together is 1.173 times larger than the area of just the rose garden, how wide will the new path be?

word problems

4.4 – The Quadratic Formula Date:

Key Ideas:

Quadratic equations can be solved by graphing (4.1), factoring (4.2), the square root method (4.3) and/or completing the square (4.3). Each of these methods have advantages and limitations.

Any quadratic equation can be solved using something called the quadratic formula. If the quadratic equation is in standard form (𝑎𝑥2 ± 𝑏𝑥 ± 𝑐 = 0), the quadratic formula is:

There could be 0, 1, or 2 resulting roots, depending on the discriminant, the expression under the square root (𝑏2 − 4𝑎𝑐). Solutions can be written in simplest radical form, or decimal form.

How would the discriminant determine the nature of the roots (the number of roots)?

If 𝑏2 − 4𝑎𝑐 > 0,

If 𝑏2 − 4𝑎𝑐 = 0,

If 𝑏2 − 4𝑎𝑐 < 0,

Example – Determine the nature of the roots, and then solve 3𝑥2 + 5𝑥 − 2 = 0 using

the quadratic formula

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

discriminant

quadratic

formula

Example – Determine the nature of the roots for 1

4𝑥2 − 3𝑥 + 9 = 0

Examples – Solve using the quadratic formula. Leave answers in exact form.

a) 𝑥2 = 2𝑥 + 1 b) 3𝑥2 − 5𝑥 = −9

Example – A picture measures 30cm by 21cm. You crop the picture by removing strips of the same width from the top and one side of the picture. This reduces the area to 40% of the original area. Determine the width of the removed strips.

word problems