Geometry (2nd semester)



Algebra 2 Name: _____________________________

Quadratics: Worksheet Quad5 Period: _____

A quadratic equation is an equation that can be written in the form [pic] where a, b and c are numbers. Some quadratic equations can be solved by factoring and using the fact that if the product of two numbers is 0, then one of the numbers must be 0:

EXAMPLE 1: Solve: [pic]

SOLUTION: First subtract x and 6 from both sides to get an expression = 0:

[pic]

Then use FOIL to factor: [pic]

Since the product of the two numbers, (2x + 3) and (x – 2), is 0 one of them must be 0, so set 2x + 3 = 0 and x – 2 = 0 and solve these: [pic] or x = 2

Not all quadratic trinomials can be factored, and you might be uncomfortable with factoring anyway. A sure method for finding the solutions (also called “roots”) of a quadratic equation is to use the quadratic formula:

QUADRATIC FORMULA: If [pic] then [pic]

EXAMPLE 2: Solve: [pic]

SOLUTION: In this equation, a = 5, b = 6, and c = –8, so plug these values in the formula:

[pic]

[pic]

There are two solutions: [pic] and [pic]

EXAMPLE 3: Solve: [pic]

SOLUTION: First expand the left side, then subtract 2 from both sides to get [pic].

This is the same as [pic] so a = 1, b = –2, and c = –2. Plug these values in the quadratic formula:

[pic]

Simplify [pic] and write as two fractions:

[pic]

So the two solutions are [pic] and [pic].

Sometimes you have to simplify an expression in an equation using FOIL. This is a method for simplifying the product of binomials. In reverse, it is also a method for factoring a quadratic expression.

EXAMPLE 4: Use FOIL to simplify [pic]

SOLUTION: FOIL stands for First, Outer, Inner, Last. These refer to products of the terms in each binomial:

(3x + – 5)(2x + 3) = [pic]

EXERCISES:

Problems 1-12: Solve the quadratic equation using factoring or the quadratic formula. Give answers in simplified, exact form.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. [pic] 12. [pic]

13. A “golden rectangle” is a rectangle which, when partitioned into a square and another rectangle,

results in the new rectangle having the same shape as the original rectangle. That is, in the figure below rectangle ABCD is a golden rectangle if rectangle BCFE is similar to it, so [pic].

Some authors make a big deal about this and claim that such a rectangle is the “most pleasing” to the eye. In fact, Dan Brown in The Da Vinci Code claims that golden rectangles and the related ratio (symbolized by the Greek letter Φ, “Phi”) occur all over the place, including most of Leonard da Vinci’s paintings. This, of course, is a lot of hooey. (See, for example, Martin Gardner’s book, Fads and Fallacies in the Name of Science, Dover, 1957).

Find the exact value and an approximate value (to 3 decimal places) of Φ.

(Hint: Let BC = 1 unit. Then AB is Φ. Maybe you would prefer to call it x.)

14. Find AB:

15. Find FG:

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The number under the square root of the quadratic formula determines the types of roots you get when a, b and c are real numbers. If that number is negative, then the roots contain the imaginary number i and are complex conjugates. If that number is 0, then there is only one root and it is a real number, and if positive there are two real roots.

So the magic number that determines the nature of the roots is [pic]. Since this is an important concept, it is given a special name: it is called the discriminant. That is,

The discriminant of the quadratic expression [pic] is [pic].

Notice that the discriminant does NOT contain the square root. It is simply the stuff inside the square root. Graphically, if the curve [pic] does not intersect the x-axis then the discriminant must be negative, if the graph is tangent to the x-axis then the discriminant is 0, and if it intersects the x-axis twice then the discriminant is positive.

More Exercises:

16. If k is a real number and the equation [pic] has no real roots, then what must the be the value of k?

17. If k is a real number and the equation [pic] has only one real root, then what must the be the value of k?

18. If k is a real number and the equation [pic] has two real roots, then what must the be the value of k?

19. If k is a real number and the equation [pic] has only one real root, then what must the be the value of k?

20. If k is a positive real number and the graph of [pic]

does not intersect the x-axis, then what must be the value of k?

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C

B

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