Quadratic Equations,

Solving a Quadratic Equation

• by factorization

• by graphical method

• by taking square roots

• by quadratic equation

By taking square roots

kx 2

kx 2

kx

In this form we could have the case where b = 0.

02 cbxaxRemember standard form for a quadratic equation is:

02 cax002 cxax

When this is the case, we get the x2 alone and then square root both sides.

062 2 x Get x2 alone by adding 6 to both sides and then dividing both sides by 2

+ 6 + 6

62 2 x2 2

32 x

Now take the square root of both sides remembering that you must consider both the positive and negative root.

3x Let's check: 0632

2 0632

2

066 066

Now take the square root of both sides remembering that you must consider both the positive and negative root.

By taking square roots

4)32( 2 x432 x

232 x52 x

5.2xA quadratic equation must contain two roots.

?

By taking square roots

4)32( 2 x

432 x

232 x

152 orx 5.05.2 orx

By taking square roots

813 2 x

183 2 x

93 2 x

3

9

3

3 2

x

32 x

32 x

3x

By taking square roots

1125 2 x

5112 2 x

62 2 x

2

6

2

2 2

x

32 x

No solution, x² cannot be negative

Exercise 9F Page 298

02 cbxaxWhat if in standard form, c = 0?

002 bxaxWe could factor by pulling an x out of each term.

032 2 xx Factor out the common x

032 xx Use the Null Factor law and set each factor = 0 and solve.

032or 0 xx

2

3or 0 xx If you put either of these values in for x

in the original equation you can see it makes a true statement.

By factorization

01072 xx0)2)(5( xx

02__05 xorx2__5 xorx

roots (solutions)

A quadratic equation is an equation equivalent to one of the form

Where a, b, and c are real numbers and a 0

02 cbxax

To solve a quadratic equation we get it in the form above and see if it will factor.

652 xx Get form above by subtracting 5x and adding 6 to both sides to get 0 on right side.

-5x + 6 -5x + 6

0652 xx Factor.

023 xx Use the Null Factor law and set each

factor = 0 and solve.

02or 03 xx 3x 2x

So if we have an equation in x and the highest power is 2, it is quadratic.

02 cbxaxWhat are we going to do if we have non-zero values for a, b and c but can't factor the left hand side?

0362 xx This will not factor so we will complete the square and apply the square root method.

First get the constant term on the other side by subtracting 3 from both sides.362 xx

___ 3___ 62 xx

Let's add 9. Right now we'll see that it works and then we'll look at how to find it.

9 9 69 62 xx

69 62 xx Now factor the left hand side.

633 xx

two identical factors

63 2 xThis can be written as:

Now we'll get rid of the square by square rooting both sides.

63 2 x Remember you need both the positive and negative root!

63 x Subtract 3 from both sides to get x alone.

63 xThese are the answers in exact form. We can put them in a calculator to get two approximate answers.

55.063 x 45.563 x

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