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Quadratic functions
A quadratic function has the general form
The quadratic function is also known as a second degree polynomial.
1. Solving quadratic equation
There are three methods of solving quadratic equations.
2. Factorization
When the factors are multiplied that is expanded, you get back the quadratic equation.
Example
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\(x^2+5x+6=0 \)
\(x+3x+2=0\)
\(x=-3 \text { and } x=-2\)
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Example
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\(2x^2-x-1=0 \)
\(x-12x+1=0 \)
\(x=-12 \text { and } x=1 \)
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3. Completing the square
For the quadratic expression
you complete the square by first factoring out a :
| \(a(x^2+{b \over a} \times x +{c \over a})\) |
and the using the formula :
| \(a[(x+{b \over 2a}^2)+{c \over a}-{b^2 \over 4a^2}]\) |
Example
For
the coefficients are a=1, b=5 and c=6. Then, using the above formula, you have
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\((x+{5 \over 2(1)})^2+6-{5^2 \over 4(1)^2}=0\)
\((x+{5 \over 2})^2+6-{25 \over 4}=0\)
\( (x+{5 \over2 })^2-{1 \over 4}=0\)
\((x+{5 \over 2})^2={1 \over 4}\)
\(x+{5 \over 2}-{1 \over 2} \text { and } x+{5 \over 2}={1 \over 2}\)
\(x=-{1 \over 2}-{5 \over 2} \text{ and } x={1 \over 2}-{5 \over 2}\)
\(x=-3 \text{ and } x=-2 \)
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Example
For
the coefficients are a=2, b=-1 and c= -1. Then, using the completing square method, you have
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\( 2(x^2-{1 \over 2}x-{1 \over 2})=0\)
\( 2[(x-{1 \over 4})^2-{1 \over 2}-{1 \over 16}]=0\)
\( 2[(x-{1 \over 4})^2-{9 \over 16}]=0\)
\( (x-{1 \over 4})^2-{9 \over 16}=0\)
\( (x-{1 \over 4})^2={9 \over 16}\)
\( x-{1 \over 4}=-{ 3 \over 4} \text { and } x-{1 \over 4}={3 \over 4} \)
\( x=-{3 \over 4}+{1 \over 4} \text { and } x={3 \over 4} +{1 \over 4}\)
\(x=-{1 \over 2} \text { and } x=1 \)
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4. Quadratic formula
For the quadratic equation
the quadratic formula to find x is :
| \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) |
Example
For
where you have seen that the coefficients are a=1, b=5 and c=6,
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\(x = {-5 \pm \sqrt{5^2-4(1)(6) }\over 2(1)}\)
\(x = {-5 \pm \sqrt{25-24} \over 2}\)
\(x = {- 5\pm \sqrt{1} \over 2}\)
\(=-{5-1 \over 2}, {-5+1 \over 2}\)
\(={-6 \over 2}, {-4 \over 2}\)
\(=-3,-2 \)
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5. Roots of quadratic equation
The number of roots and the type of roots of the quadratic equation
can be determined by the discriminant D=b2-4ac.
- If D>0, the quadratic equation has two different real roots
- If D=0, the quadratic equation has one real root, repeated twice
- If D<0, the quadratic equation has two different complex roots
Example
Notice that in
given above,
and so you had two different real roots x=-3 and x=-2.
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