This document discusses different methods for solving quadratic equations: 1) The quadratic formula, which can be used to solve any quadratic equation in the form ax^2 + bx + c = 0. 2) Factoring, which involves breaking the equation into factors that can be multiplied together. This works if the leading coefficient is 1. 3) Graphing the equation to find where the graph intersects the x-axis.

1. Different Ways of Solving a
Quadratic Equation

2. The quadratic formula
The quadratic formula is an important method of solving a quadratic equation
because it works in any scenario, as long as the form of the equation is ax^2 + bx
+ c = 0
The quadratic formula can be written as:
x =
-b±√(b2-4ac)
2a

3. The quadratic formula
For reference to what a, b, and c are, the standard form of a quadratic equation
can be written as ax^2 + bx + c = 0
You would plug in these values of a, b, and c into the equation and you would
get your 2 solutions. Let’s do an example in the next slide

4. The quadratic formula
Let’s say we have x2 – 10x + 9. We
know that a = 1, b = -10, c = 9
Let’s now plug in these values into the
quadratic formula:
x =
-(-10)±√((-10)2-4(1)(9))
2(1)
When we evaluate everything, we
get x = 5 ± 4. So as a result, one of
our solutions is 9 and the other
solution is 1. Plug them both into
the equation and verify that both x
values are correct
x = 5 ± 4
x = 9
x = 1

5. Factoring
Another popular method of solving quadratic equations is factoring – which is
essentially breaking up the equation into smaller pieces which can be
multiplied to equal the whole
If we have an equation in standard form and the leading coefficient (a) is 1, there
is a very easy way to factor
ax2 + bx + c

6. Factoring
When factoring a quadratic equation in the form of ax^2 + bx + c, where a is 1,
you look for two numbers (let's call them m and n) that satisfy the following
conditions: the product of m and n equals c and the sum of m and n equals b
Let’s refer to our old example of x2 – 10x + 9
ax2 + bx + c 1x2-10x+9

7. Factoring
The products of m and n: m * n = 1 * 9 = 9. The sum of m and n: m + n = -1 + (-9) = -
10. Now, we need to find two numbers that satisfy these conditions. The
numbers that fulfill this are -1 and -9, as: -1 * -9 = 9, -1 + (-9) = -10
How would we write this out in factored form?
Factored form is (x – b)(x – c). b and c are the x values that we just found
Meaning that if we had 2 negative numbers, you would subtract the negative
numbers in both cases, which make them positive
(x – (-9))(x – (-1)) becomes (x + 9)(x + 1)

8. Factoring
Let’s say we have: 2x2 + 3x – 14 a = 2 b = 3 c = -14
If the leading coefficient is NOT 1, factoring becomes a lot more complicated,
and we use a method called Factoring by Grouping
Let’s showcase this through an example:
We would first multiply a and b, and find two
numbers that multiply to get -28 and add to get b
(3)
In our case, 2 * -14 is -28. Two numbers that multiply
to get -28 and add to get 3 are 7 and -4
2 * -14 = -28

9. Factoring
To identify these values faster, we suggest writing down all the factors and
choosing which 2 meet the criteria
You would then rewrite the equation like this:
2x2+ 7x – 4x - 14 (2x2 + 7x) + (– 4x - 14)
You factor each bit
separately:
x(2x + 7) -2(2x + 7)
You can see that (2x+7) is repeated in the final result. Eliminate one and multiply
it by the other values, x and -2. Thus, you will get (x - 2)(2x + 7)
You have now reached factored form

10. Graphing
Lastly, the easiest most foolproof
method is to solve by graphing
Graph your equation, see where it
touches the x axis, and those are
the solutions
X2 + 3x – 4

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