5.6
The Quadratic Formula
and the Discriminant
In this lesson we will learn how to use the Quadratic Formula to solve quadratic equations.
We will also look at the value of the Discriminant to determine how many solutions and what type of roots they 1
To solve and equation in the form
ax2 + bx + c = 0 where a ≠ 0
we can use the QUADRATIC FORMULA
to get the solutions for x
x = ­ b ±
2
b ­ 4ac
2a
1. Make sure the equation is equal to zero and in the form: ax2 + bx + c = 0
2. Find a, b, and c and plug them into Quadratic Formula
*Be sure to use ( ) each time you replace a variable with a value
3. Solve for x and check to see if you can simplify and reduce any radical expressions or fractions
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Example 1: Solve using the Quadratic Formula
4x2 ­ 8x + 1 =0
2 real; x = 1.87 or .133
OR x = (2±√(3))/2
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One more....
Example 2: Solve using the Quadratic Formula
x2 ­ 16x = ­64
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These are the following types of solutions you can end up with for a quadratic equation
ONE real TWO real We can use the value of the DISCRIMINANT to help us predict how many and what kind of zeros/roots the quadratic function has.
x = ­ b ±
2
b ­ 4ac
2a
TWO Imaginary/Complex
If the value of the discriminant is:
POSITIVE: 2 real zeros
NEGATIVE: 2 imaginary/complex zeros (contain i)
= 0 : 1 real zero
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Example 3: Find the value of the discriminant and determine how many and what type of roots the function has.
7x2 ­ 11x = ­5
2 imaginary; discriminant is ­19
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Here are the ways we've learned how to solve quadratic functions and the Quadratic Formula is great because it can be used to solve ANY quadratic equation.
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homework:
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