Methods for solving a quadratic function

In this article, we will learn the three most common ways to solve a quadratic function easily and quickly.

Trinomial
Quadratic Formula
Completing the Square

Reminder:

The basic quadratic function equation is:
$Y=ax^2+bx+c$

When:
$a$ - the coefficient of $X^2$
$b$ - the coefficient of $X$
$c$ - the constant term

$a$ must be different from $0$
$b$ or $c$ can be $0$
$a,b,c$ can be negative/positive
The quadratic function can also look like this:
- $Y=ax^2$
- $Y=ax^2+bx$
- $Y=ax^2+c$

Detailed explanation

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Test yourself on solution of quadratic equation!

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term

What is the value of $$ c $$ in the function $$ y=-x^2+25x $$ ?

Practice more now

How do you solve a quadratic function using trinomial?

$Y=ax^2+bx+c$
Find $2$ numbers that satisfy the following two conditions:

$first~number \cdot second~number = c \cdot a$
$first~number + second~number = b$

How do we proceed?
First of all, let's write down on the side:

*Relevant notation in the Word file*

Then–

We will find all the numbers whose product is $a\cdot c$ and write them down.
We will check which pair of numbers from the pairs we found earlier will give us the sum $b$ .
We will write the pair of numbers that meets the $2$ conditions in this form:
$0= (x+one~solution)(x+second~solution)$
or with subtraction operations.
$2$ solutions of the quadratic equation will be those that solve the equation above (we will reverse the sign for the pair of numbers we found)

Tip – It is recommended to use the trinomial method when $a=1$

For more reading and practice on trinomials, click here!

And now let's practice!
Solve the quadratic function in front of you using trinomial:
$x^2-9x+14$

Solution:
First of all, let's write down on the side:

*Relevant notation in the Word file*

Let's find all the numbers whose product is $14$ (and remember the negative numbers as well)
We get:
$14,1$
$7,2$
$-14, -1$
$-7 ,-2$

Now, let's check which pair of numbers from the pairs we found earlier will give us the sum (-9)

*Relevant notation in the Word file*

The pair of numbers that managed to meet both conditions is $-7 ,-2$

We write the factorization:
$(x-7)(x-2)=0$
The solutions:
$X=7$
$X=2$

How do you solve a quadratic function using the quadratic formula?

$Y=ax^2+bx+c$

Meet the quadratic formula:
$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$

All you need to do is arrange the parameters of the quadratic function, substitute into the equation once with a plus and once with a minus, and find the solutions.

To learn more about the quadratic formula, click here!

Let's practice!
In front of us is the quadratic function:
$2x^2-3x+1$

Let's solve it using the quadratic formula:
First, let's arrange the parameters:
$a=2$
$b=-3$
$c=1$

Now, we substitute into the quadratic formula:
For the first time with plus-

$\frac{-(-3)+\sqrt{(-3)^2-4*2*1}}{2*2}=$

$=\frac{3+\sqrt1}4=\frac{4}4=1$

The second time with minus:

$\frac{-(-3)-\sqrt{(-3)^2-4*2*1}}{2*2}=$

$=\frac{3-\sqrt1}4=\frac{2}4=\frac{1}2$

We got $2$ solutions –
$X=\frac{1}2,1$

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Test your knowledge

Question 1

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of $$ b $$ in this quadratic equation:

$$ y=4x^2-16 $$

Question 2

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of $$ c $$ in this quadratic equation:

$$ y=5+3x^2 $$

Question 3

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ b $$ in the equation

$$ y=3x^2+10-x $$

How do you solve a quadratic function by completing the square?

To use the method of completing the square, let's recall some of the formulas for the shortened multiplication:

$a^2+2ab+b^2=(a+b)^2$

$a^2-2ab+b^2=(a-b)^2$

Solution method with example:
Here is the function $4x^2+8x-5$

Let's look at the quadratic function and focus only on $ax^2+bx$ .
For now, we'll ignore $c$ . In the example, we'll focus on $4x^2+8x$
Let's recall the formulas for completing the square and ask, what expression can we put inside the parentheses squared, that is, which $(a-b)^2$ or $(a+b)^2$ as relevant, that will give us what appears in the pair we focused on $ax^2+bx$

In the example-
The appropriate formula for completing the square is:
$a^2+2ab+b^2=(a+b)^2$

Let's ask, what should we put as $a$ and $b$ to get $4x^2+8x$ ?
The answer is - $(2x+2)^2$
Let's expand this expression according to the formula for completing the square and get:
$4x^2+8x+4$

Note that the expression in parentheses also brings with it a certain number and not just the pair we focused on, so we will need to cancel it.
If the added number is negative, we will add it to the equation to cancel it, and if the number is positive, we will subtract it from the equation to cancel it.
Additionally, we will return to $C$ from the original function and write it in the equation as well.

In the example:
$4x^2+8x+4$
The number $4$ was added. To cancel it, we will subtract $4$ and not forget the original $C$
$4x^2+8x-4-5=$
We will substitute the pair $ax^2+bx$ with the appropriate expression in parentheses squared that we found and arrange the equation – we will get the completion of the square.

In the example:
$(2X+2)^2-4-5=$
$(2x+2)^2-9$

Now let's set the equation to 0 and solve:
$(2x+2)^2-9=0$

$(2x+2)^2=9$

$(2x+2)^2=3^2$
and also
$(2x+2)^2=(-3)^2$

Solution 1:
$2x+2=3$
$2x=1$
$x=\frac{1}2$

Solution 2:
$2x+2=-3$
$2x=-5$
$x=-2\frac{1}2$

For more information on completing the square, click here!

Do you know what the answer is?

Question 1

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

Question 2

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of $$ c $$ in this quadratic equation:

$$ y=-5x^2+4x-3 $$

Question 3

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of $$ b $$ in the equation

$$ y=2x-3x^2+1 $$

examples with solutions for solution of quadratic equation

Exercise #1

Solve the following equation:

$x^2+5x+4=0$

Video Solution

Step-by-Step Solution

The parameters are expressed in the quadratic equation as follows:

aX2+bX+c=0

We substitute into the formula:

-5±√(5²-4*1*4)
2

-5±√(25-16)
2

-5±√9
2

-5±3
2

The symbol ± means that we have to solve this part twice, once with a plus and a second time with a minus,

This is how we later get two results.

-5-3 = -8
-8/2 = -4

-5+3 = -2
-2/2 = -1

And thus we find out that X = -1, -4

Answer

$x_1=-1$ $x_2=-4$

Exercise #2

$x^2+9=0$

Solve the equation

Video Solution

Step-by-Step Solution

The parameters are expressed in the quadratic equation as follows:

aX2+bX+c=0

We identify that we have:
a=1
b=0
c=9

We recall the root formula:

$Roots formula | The version$

We replace according to the formula:

-0 ± √(0²-4*1*9)

We will focus on the part inside the square root (also called delta)

√(0-4*1*9)

√(0-36)

√-36

It is not possible to take the square root of a negative number.

And so the question has no solution.

Answer

No solution

Exercise #3

$60-16y+y^2=-4$

Video Solution

Step-by-Step Solution

Let's solve the given equation:

$60-16y+y^2=-4$ First, let's arrange the equation by moving terms:

$60-16y+y^2=-4 \\ 60-16y+y^2+4=0 \\ y^2-16y+64=0$ Now, let's note that we can break down the expression on the left side using the short quadratic factoring formula:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$ This is done using the fact that:

$64=8^2$ So let's present the outer term on the right as a square:

$y^2-16y+64=0 \\ \downarrow\\ \textcolor{red}{y}^2-16y+\textcolor{blue}{8}^2=0$ Now let's examine again the short factoring formula we mentioned earlier:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$ And the expression on the left side of the equation we got in the last step:

$\textcolor{red}{y}^2-\underline{16y}+\textcolor{blue}{8}^2=0$ Let's note that the terms $\textcolor{red}{y}^2,\hspace{6pt}\textcolor{blue}{8}^2$ indeed match the form of the first and third terms in the short multiplication formula (which are highlighted in red and blue),

But in order for us to break down the relevant expression (which is on the left side of the equation) using the short formula we mentioned, the match to the short formula must also apply to the remaining term, meaning the middle term in the expression (underlined):

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$ In other words - we'll ask if it's possible to present the expression on the left side of the equation as:

$\textcolor{red}{y}^2-\underline{16y}+\textcolor{blue}{8}^2 =0 \\ \updownarrow\text{?}\\ \textcolor{red}{y}^2-\underline{2\cdot\textcolor{red}{y}\cdot\textcolor{blue}{8}}+\textcolor{blue}{8}^2 =0$ And indeed it holds that:

$2\cdot y\cdot8=16y$ So we can present the expression on the left side of the given equation as a difference of two squares:

$\textcolor{red}{y}^2-2\cdot\textcolor{red}{y}\cdot\textcolor{blue}{8}+\textcolor{blue}{8}^2=0 \\ \downarrow\\ (\textcolor{red}{y}-\textcolor{blue}{8})^2=0$ From here we can take out square roots for the two sides of the equation (remember that there are two possibilities - positive and negative when taking out square roots), we'll solve it easily by isolating the variable on one side:

$(y-8)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ y-8=\pm0\\ y-8=0\\ \boxed{y=8}$

Let's summarize then the solution of the equation:

$60-16y+y^2=-4 \\ y^2-16y+64=0 \\ \downarrow\\ \textcolor{red}{y}^2-2\cdot\textcolor{red}{y}\cdot\textcolor{blue}{8}+\textcolor{blue}{8}^2=0 \\ \downarrow\\ (\textcolor{red}{y}-\textcolor{blue}{8})^2=0 \\ \downarrow\\ y-8=0\\ \downarrow\\ \boxed{y=8}$

So the correct answer is answer a.

Answer

$y=8$

Exercise #4

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term

What is the value of $c$ in the function $y=-x^2+25x$ ?

Video Solution

Answer

$c=0$

Exercise #5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of $b$ in this quadratic equation:

$y=4x^2-16$

Video Solution

Answer

$b=0$

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