Concept

Quadratic Equation

A quadratic equation is a polynomial equation of second degree, meaning the highest exponent of its monomials is $2 .$ A quadratic equation can be written in standard form as follows.

$a x^{2} + b x + c = 0$

Here, the leading coefficient

a

is non-zero, which guarantees that the

x^{2} -

term is present. Solving a quadratic equation means finding the zeros of the related quadratic function. Therefore, quadratic equations can have at most two solutions.

Quadratic Equation a x^{2} + b x + c = 0 Related Function y = a x^{2} + b x + c

This type of equation can be solved using several methods, such as graphing and factoring.

Method

Solving a Quadratic Equation Graphically

A quadratic equation can be solved graphically by drawing the parabola that corresponds to the related quadratic function and identifying its zeros. Consider the following equation as an example.

2 x^{2} + 3 x - 5 = 5 x + 7

These four steps can be followed to find the solutions of the equation.

1

Write the Equation in Standard Form

If neither side of the equation equals

0,

rearrange it so that all terms are on the same side.

2 x^{2} + 3 x - 5 = 5 x + 7 ⇕ 2 x^{2} + 3 x - 5 - 5 x - 7 = 0

If there is more than one term with the same degree, simplify the equation by combining like terms.

2 x^{2} + 3 x - 5 - 5 x - 7 = 0 ⇕ 2 x^{2} - 2 x - 12 = 0

The equation is in standard form now.

2

Graph the Related Function

The side with the variables can now be seen as a function

f (x) .

Quadratic Equation 2 x^{2} - 2 x - 12 = 0 Related Function f (x) = 2 x^{2} - 2 x - 12

To graph the quadratic function in standard form, its characteristics should be identified first.

$f (x) = 2 x^{2} - 2 x - 12$
Direction	Vertex	Axis of Symmetry	$y -$ intercept
$a > 0 \Rightarrow$ upward	$(\frac{1}{2}, - \frac{2 5}{2})$	$x = \frac{1}{2}$	$(0, - 12)$

In addition to these points, the reflection of the $y -$ intercept across the axis of symmetry is at $(1, - 12),$ which is also on the parabola. Now, the graph can be drawn.

3

Identify Any Points With

y -

coordinate

0

Now, find all the points on the graph that have a $y -$ coordinate equal to $0 .$

4

Identify the

x -

coordinates of Those Points

The $x -$ coordinates of the identified points solve the equation $f (x) = 0 .$

In the example, the

x -

coordinates of the points where the curve intercepts the

x -

axis are

- 2

and

3 .

Note that solving an equation graphically does not necessarily lead to an exact answer. To verify a solution, substitute it into

f (x)

and evaluate the expression. Verifying the solution

x = - 2

is done by evaluating

f (- 2) .

f (x) = 2 x^{2} - 2 x - 12

Substitute

$x = - 2$

f (- 2) = 2 (- 2)^{2} - 2 (- 2) - 12

Evaluate right-hand side

CalcPow

Calculate power

f (- 2) = 2 (4) - 2 (- 2) - 12

Multiply

f (- 2) = 8 - 2 (- 2) - 12

MultPosNeg

$a (- b) = - a \cdot b$

f (- 2) = 8 - (- 4) - 12

SubNeg

$a - (- b) = a + b$

f (- 2) = 8 + 4 - 12

AddSubTerms

Add and subtract terms

f (- 2) = 0

When

x = - 2,

the value of

f (x)

is

0 .

Therefore,

x = - 2

is an exact solution. The solution

x = 3

can now be verified by following the same procedure.

Solution	Substitute	Evaluate
$x = - 2$	$f (- 2) = 2 (- 2)^{2} - 2 (- 2) - 12$	$f (- 2) = 0 ✓$
$x = 3$	$f (3) = 2 (3)^{2} - 2 (3) - 12$	$f (3) = 0 ✓$

Since

f (3) = 0,

x = 3

is also a solution. Therefore, the equation

2 x^{2} + 3 x - 5 = 5 x + 7

has two solutions.

x = - 2 and x = 3

Theory

Simple Quadratic Equation

Simple quadratic equations take the form

a x^{2} + c = 0

and are solved using inverse operations. Once

x^{2}

remains on the left-hand side, the equation can be written as

x^{2} = d,

where

d = \frac{- c}{a}

. The value of

d

gives the number of solutions the equation has.

d > 0 d = 0 d < 0 : 2 real solutions : 1 real solution : 0 real solutions

Method

Solving Quadratic Equations with Square Roots

Simple quadratic equations are quadratic equations whose linear coefficient

b

is equal to

0 .

a x^{2} + c = 0

This type of equation can be solved using inverse operations, and two steps must be followed.

Isolate $x^{2} .$
Take square roots on both sides of the equation.

As an example, consider the equation

5 x^{2} - 500 = 0 .

1

Isolate

x^{2}

Inverse operations will be used to isolate

x^{2} .

Here,

500

will be added to both sides of the equation. Then, the left- and right-hand sides will be divided by

5 .

5 x^{2} - 500 = 0

AddEqn

$LHS + 500 = RHS + 500$

5 x^{2} = 500

DivEqn

$LHS / 5 = RHS / 5$

x^{2} = 100

2

Take Square Roots

Now that

x^{2}

has been isolated, it is necessary to undo the exponent. Exponents and radicals of the same index undo each other. Therefore, square roots undo powers of

2 .

x^{2} = 100

SqrtEqn

$LHS = RHS$

x^{2} = 100

$a^{2} = \pm a$

x = \pm 100

CalcRoot

Calculate root

x = \pm 10

StateSol

State solutions

x = 10 x = - 10

Note that the negative solution is also considered along with the principal root when solving the equation.

Method

Factoring a Quadratic Trinomial

When trying to factor a quadratic trinomial of the form

a x^{2} + b x + c,

it can be difficult to see its factors. Consider the following expression.

8 x^{2} + 26 x + 6

Here,

a = 8,

b = 26,

and

c = 6 .

There are six steps to factor this trinomial.

1

Factor Out the GCF of

a,

b,

and

c

To fully factor a quadratic trinomial, the Greatest Common Factor (GCF) of

a,

b,

and

c

has to be factored out first. To identify the GCF of these numbers, their prime factors will be listed.

8 = 26 = 6 = 2 \cdot 2 \cdot 2 2 \cdot 13 2 \cdot 3

It can be seen that

8,

26,

and

6

share exactly one factor,

2 .

GCF (8, 26, 6) = 2

Now,

2

can be factored out.

8 x^{2} + 26 x + 6

SplitIntoFactors

Split into factors

2 (4) x^{2} + 2 (13) x + 2 (3)

FactorOut

Factor out $2$

2 (4 x^{2} + 13 x + 3)

In the remaining steps, the factored coefficient

2

before the parentheses can be ignored. The new considered quadratic trinomial is

4 x^{2} + 13 x + 3 .

Therefore, the current values of

a,

b,

and

c,

are

4,

13,

and

3,

respectively. If the GCF of the coefficients is

1,

this step can be ignored.

2

Find the Factor Pair of

a c

Whose Sum Is

b

It is known that $a = 4$ and $c = 3,$ so $a c = 12 > 0 .$ Therefore, the factors must have the same sign. Also, $b = 13 .$ Since the sum of the factors is positive and they must have the same sign, both factors must be positive. All positive factor pairs of $12$ can now be listed and their sums checked.

Factors of $a c$	Sum of Factors
$1$ and $12$	$1 + 12 = 13 ✓$
$2$ and $6$	$2 + 6 = 8 \times$
$3$ and $4$	$3 + 4 = 7 \times$

In this case, the correct factor pair is $1$ and $12 .$ The following table sums up how to determine the signs of the factors based on the values of $a c$ and $b .$

$a c$	$b$	Factors
Positive	Positive	Both positive
Positive	Negative	Both negative
Negative	Positive	One positive and one negative. The absolute value of the positive factor is greater.
Negative	Negative	One positive and one negative. The absolute value of the negative factor is greater.

Such analysis makes the list of possible factor pairs shorter.

3

Write

b x

as a Sum

The factor pair obtained in the previous step will be used to rewrite the

x -

term — the linear term — of the quadratic trinomial as a sum. Remember that the factors are

1

and

12 .

13 x

WriteSum

Write as a sum

12 x + 1 x

IdPropMult

Identity Property of Multiplication

12 x + x

The linear term

13 x

can be rewritten in the original expression as

12 x + x .

4 x^{2} + 13 x + 3 ⇕ 4 x^{2} + 12 x + x + 3

4

Factor Out the GCF of the First Two Terms

The expression has four terms, which can be grouped into the

first

two

and the

last

two

terms .

Then, the GCF of each group can be factored out.

4 x^{2} + 12 x + x + 3

The first two terms,

4 x^{2}

and

12 x,

can be factored.

4 x^{2} = 12 x = 2 \cdot 2 \cdot x \cdot x 2 \cdot 2 \cdot 3 \cdot x

The GCF of

4 x^{2}

and

12 x

is

2 \cdot 2 \cdot x = 4 x .

4 x^{2} + 12 x + x + 3

Rewrite

Rewrite $4 x^{2}$ as $4 x \cdot x$

4 x \cdot x + 12 x + x + 3

Rewrite

Rewrite $12 x$ as $4 x \cdot 3$

4 x \cdot x + 4 x \cdot 3 + x + 3

FactorOut

Factor out $4 x$

4 x (x + 3) + x + 3

5

Factor Out the GCF of the Last Two Terms

The process used in Step

3

will be repeated for the last two terms. In this case,

x

and

3

cannot be factored, so their GCF is

1 .

4 x (x + 3) + x + 3

Rewrite

Rewrite $x$ as $1 \cdot x$

4 x (x + 3) + 1 \cdot x + 3

Rewrite

Rewrite $3$ as $1 \cdot 3$

4 x (x + 3) + 1 \cdot x + 1 \cdot 3

FactorOut

Factor out $1$

4 x (x + 3) + 1 (x + 3)

6

Factor Out the Common Factor

If all the previous steps have been performed correctly, there should now be two terms with a common factor.

4 x (x + 3) + 1 (x + 3)

The common factor will be factored out.

4 x (x + 3) + 1 (x + 3)

FactorOut

Factor out $(x + 3)$

(4 x + 1) (x + 3)

The factored form of

4 x^{2} + 13 x + 3

is

(4 x + 1) (x + 3) .

Remember that the original trinomial was

8 x^{2} + 26 x + 6

and that the GCF

2

was factored out in Step

1 .

This GCF has to be included in the final result.

8 x^{2} + 26 x + 6 = 2 (4 x + 1) (x + 3)

Solve the quadratic equation by factoring

fullscreen

Solve the following equation by factoring.

x^{2} - 2 x - 3 = 0

Show Solution

We'll focus on the left-hand side of the equation before solving for

x .

Notice that

1

and

- 3

are the factors that multiply to equal

- 3

and add to

- 2 .

x^{2} - 2 x - 3 = 0

Rewrite

Rewrite $- 2 x$ as $x - 3 x$

x^{2} + x - 3 x - 3 = 0

FactorOut

Factor out $x$

x (x + 1) - 3 x - 3 = 0

FactorOut

Factor out $- 3$

x (x + 1) - 3 (x + 1) = 0

FactorOut

Factor out $(x + 1)$

(x + 1) (x - 3) = 0

Notice that, in factored form, the numbers in the parentheses with

x

are the numbers from the chosen factor pair. This is because the coefficient of

x^{2}

is

1 .

We could have written the expression in factored form immediately after finding the factor pair. Lastly, we can use the zero product property to solve the equation.

x + 1 = 0 x - 3 = 0 \leftrightarrow x = - 1 \leftrightarrow x = - 3

Thus, the solutions to the equation

x^{2} - 2 x - 3 = 0

are

x = - 1

and

x = 3 .

Solving Quadratic Equations

Channels

Direct messages

Concept

Quadratic Equation

Method

Solving a Quadratic Equation Graphically

Theory

Simple Quadratic Equation

Method

Solving Quadratic Equations with Square Roots

Method

Factoring a Quadratic Trinomial

Example

Solve the quadratic equation by factoring