1. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
Standard form of a quadratic equation is
ax² + bx + c = 0
iff a ≠ 0, where a, b, and c are real numbers,
b and/or c can be equal zero
a is the coefficient of the x² term
b is the coefficient of the x term
c is the constant (the number without a variable)
the coefficient is the number, sign included in front of the variable

2. How to Solve a Quadratic Equation
Make sure your equation is in Standard Form.
every term to the LHS
Find the values of a, b, and c.
a = , b = , c =
Does the parabola open up or down?

3. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
This is a “plug and chug” problem.
Just follow the steps.
You plug in the numbers for the variables and do the operations.
Let’s try this one x² - 8x + 15 = 0
a =
b =
c =

4. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
Did you get these values?
x² - 8x + 15 = 0
a = 1
b = -8
c = 15

5. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
Next step is to find the axis of symmetry.
this is the vertical line that “cuts” the parabola into
two symmetrical halves
x² - 8x + 15 = 0
a = 1 Use this to find the axis of symmetry
b = x = - b
c = a

6. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
This is another “plug and chug” problem.
You plug in the numbers for the variables and do the operations.
x² - 8x + 15 = 0
Did you find the axis of symmetry?
x = - b x = - (-8)
2a (1)
x = 8 or x = 4 when simplified 2

7. How to Solve a Quadratic Equation
Make sure your equation is in Standard Form.
Find the values of a, b, and c.
Does the parabola open up or down?
Calculate the axis of symmetry.

8. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
x² - 8x + 15 = 0
The axis of symmetry is x = 4
Now we have to find the coordinates of the vertex.
We already know the x coordinate from the axis of symmetry.
Using the corresponding equation y = x² - 8x + 15
This is another “plug and chug” problem.
You plug in the found value of x into the
original equation and do the operations to solve for y.

9. Ugh … are you having fun yet?
I love this stuff!

10. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
You plug in the found value of x into the
original equation and do the operations to solve for y.
x = 4
y = x² - 8x + 15
y = (4)² - 8(4) + 15
y =
y =
y = - 1
The coordinate of our vertex is (4,-1)

11. How to Solve a Quadratic Equation
Make sure your equation is in Standard Form.
Find the values of a, b, and c.
Does the parabola open up or down?
Calculate the axis of symmetry.
Plug that value back into the original equation to find the coordinates of the vertex.
Draw the axis of symmetry.
Plot and label the vertex.

12. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
Next is to find where the roots are.
That means where the “arms” of the parabola hit the x–axis.
x² - 8x + 15 = 0
a = 1 plug and chug these values into the above equation
b = b ± √ b² - 4ac
c = a

13. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
x² - 8x + 15 = 0
a = 1 plug and chug the values into the above equation
b = (-8) ± √ (-8)² - 4(1)(15)
c = (1)
8 ± √ (64) ± √
this leads us to 2 separate equations 8 ±

14. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
Breaking this into the two equations 8 ±
We get
2 2
5 and 3

15. How to Solve a Quadratic Equation
Make sure your equation is in Standard Form.
Find the values of a, b, and c.
Does the parabola open up or down?
Calculate the axis of symmetry.
Plug that value back into the original equation to find the coordinates of the vertex.
Calculate the “roots” of the parabola using the quadratic equation.
Draw the axis of symmetry.
Plot and label the vertex.
Plot the roots.
Draw and label the parabola.

16. The Quadratic Equation
x = -b ± √ b² - 4ac 2a
Let’s try this one x² + 3x + 2 = 0
a =
b =
c =

17. How to Solve a Quadratic Equation
Make sure your equation is in Standard Form.
Find the values of a, b, and c.
Does the parabola open up or down?
Calculate the axis of symmetry.
Plug that value back into the original equation to find the coordinates of the vertex.
Calculate the “roots” of the parabola using the quadratic equation.
Draw the axis of symmetry.
Plot and label the vertex.
Plot the roots.
Draw and label the parabola.