5.3.1
, ~5.3.1 - The quadratic
formula~
2
−𝑏± 𝑏 −4𝑎𝑐
The quadratic formula: 𝑥 = 2𝑎
Here are the different kinds of equations you can and cannot solve
with the quadratic formula:
Tip Can be solved Cannot be solved Explanation
with quadratic with quadratic
formula formula.
There can be no 2
𝑥 + 2𝑥 + 1 = 0
3 2
𝑥 + 2𝑥 + 1 = 0 This cannot be
term whose solved because
degree is higher there is a term
than 2. that is higher
2
than 𝑥 .
The coefficient 2
1𝑥 + 2𝑥 + 1 = 0 3𝑥 + 1 = 0 2
There is no 𝑥
2
of the 𝑥 term. 𝑎, term, so the
can’t be 0. equation cannot
be solved with
the quadratic
formula.
The right-hand 2
2𝑥 − 10𝑥 + 8 = 0
2
5𝑥 − 17𝑥 = 12 You can’t use
side of the the formula
equation must be until the
0. right-hand side
of the equation
is 0.
Here’s an example of how you solve an equation with the quadratic
formula. A helpful step to take before substituting an equation
(in ax^2 + bx + c = 0 format), state the value of a, b, and c. Here is
an example of how to solve a quadratic equation using the formula. The
equation in the example is -3x^2 + 5x + 6 = 0.
,An important thing to note is that when you have an equation that you
want to solve with the quadratic formula and the right-hand side of
the equation is not equal to 0, add/subtract the term for bot sides
(basically move the term to the other side) to make the right hand
side of the equation equal to 0.
, When solving an equation using the quadratic formula, if there is a
square root, you can simplify; always remember to simplify it.
Discriminant: The number under the radical sign in the quadratic
formula. It is given by the expression b2 – 4ac. The discriminant
tells you how many real solutions the equation has.
Ex: For the equation x2 + 6x + 3 = 0, a = 1, b = 6, and c = 3, so
the discriminant b2 – 4ac = 36 – 4(1)(3) = 24.
If the discriminant is positive, it has 2 solutions; if the
discriminant is equal to 0, it has one solution, and if the
discriminant is negative, it has no solutions.
How to derive the quadratic formula
, ~5.3.1 - The quadratic
formula~
2
−𝑏± 𝑏 −4𝑎𝑐
The quadratic formula: 𝑥 = 2𝑎
Here are the different kinds of equations you can and cannot solve
with the quadratic formula:
Tip Can be solved Cannot be solved Explanation
with quadratic with quadratic
formula formula.
There can be no 2
𝑥 + 2𝑥 + 1 = 0
3 2
𝑥 + 2𝑥 + 1 = 0 This cannot be
term whose solved because
degree is higher there is a term
than 2. that is higher
2
than 𝑥 .
The coefficient 2
1𝑥 + 2𝑥 + 1 = 0 3𝑥 + 1 = 0 2
There is no 𝑥
2
of the 𝑥 term. 𝑎, term, so the
can’t be 0. equation cannot
be solved with
the quadratic
formula.
The right-hand 2
2𝑥 − 10𝑥 + 8 = 0
2
5𝑥 − 17𝑥 = 12 You can’t use
side of the the formula
equation must be until the
0. right-hand side
of the equation
is 0.
Here’s an example of how you solve an equation with the quadratic
formula. A helpful step to take before substituting an equation
(in ax^2 + bx + c = 0 format), state the value of a, b, and c. Here is
an example of how to solve a quadratic equation using the formula. The
equation in the example is -3x^2 + 5x + 6 = 0.
,An important thing to note is that when you have an equation that you
want to solve with the quadratic formula and the right-hand side of
the equation is not equal to 0, add/subtract the term for bot sides
(basically move the term to the other side) to make the right hand
side of the equation equal to 0.
, When solving an equation using the quadratic formula, if there is a
square root, you can simplify; always remember to simplify it.
Discriminant: The number under the radical sign in the quadratic
formula. It is given by the expression b2 – 4ac. The discriminant
tells you how many real solutions the equation has.
Ex: For the equation x2 + 6x + 3 = 0, a = 1, b = 6, and c = 3, so
the discriminant b2 – 4ac = 36 – 4(1)(3) = 24.
If the discriminant is positive, it has 2 solutions; if the
discriminant is equal to 0, it has one solution, and if the
discriminant is negative, it has no solutions.
How to derive the quadratic formula
