# Quadratic Formula

**Warm-up**

1. A line parallel to the x-axis 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 4-5.

4-5 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.

**Warm-up**

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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