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Quadratic Formula
Warmup
1. A line parallel to the xaxis intersects a parabola
exactly once, at the point (1, 1).
What special point on the parabola is (1, 1)? It is the vertex.
2. Find the vertex and state if it is a maximum or minimum: y = 3x^{2}  6x + 7 (1,4), minimum
3. State how to translate the graph of y = 0.5x^{2} to produce y = 0.5(x  1)^{2} + 5. Right 1, up 5
4. Solve:
5. Solve:
Today we will:
1. Use the quadratic formula to solve quadratic equations
Tomorrow we will:
1. Continue with section 45.
45 The Quadratic Formula
You can find exact solutions to some quadratic equations by factoring .
However, not all quadratic equations are factorable , so we use the quadratic
formula.
The quadratic formula will enable you to solve any quadratic equation – it
always works!
The Quadratic Formula
For any quadratic equation in standard form ax^{2} + bx + c = 0 , the exact
solution (roots) are
given by:
What can we tell about a graph in standard form from the quadratic formula?
Remember to put the equation in standard form to solve. ax^{2} + bx + c
Example 1
Solve x + 4 = x^{2}
Solution
Put in ax^{2} + bx + c = 0 first!
x^{2}  x  4 = 0 ← Get the equation in standard form…
So…
a = 1, b = 1, and c = 4
Example 2
Solve 5x^{2} + 6x +1 = 0
Solution
5x^{2} + 6x +1 = 0
The equation is in standard form…
So…
a = 5, b = 6, and c = 1
Example 3
Solve x^{2}  4x + 3 = 0
Solution
The equation is in standard form…
So…
a = 1, b = 4, and c = 3
Example 4
Solve using cross products and the quadratic formula
Solution
So …
a = 1; b = 7; c = 49
Example 5
Solve using the quadratic formula: 2(x 1)^{2} + 5 = 6
Solution
We must rewrite the equation in standard form…
So… a = 2, b = 4, and c = 1 ← plug these values into the quadratic formula.
Example 6
The area of a rectangle is 25m^{2}. The rectangle is 4m longer than it is wide.
What are the
dimensions of the rectangle?
Solution
Write an equation
Let x = width
Length = x + 4 (The rectangle is 4m longer than wide)
Length and width must be positive , so x = 3.39.
Warmup
Solve each quadratic equation
1. x^{2}  4x + 3 = 0
2. x^{2} + 5x 14 = 0
3. 2x^{2} + x  3 = 0
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