Quadratic standard form to factored form calculator

Q: Can Photomath do factoring?

Just because math is getting more complicated doesnt mean we cant make it a little easier for ourselves! Factoring is one way we can make our calculations a little clearer and easier to follow.

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Table of Contents Show

Factor expressions step by step
What you will learn in this mini lesson
How do I Factor Quadratic Equations?
What if the leading coefficient A is not 1 ?
Sample Problem: How to Factor a Quadratic Equation?
How do you solve a quadratic equation in factored form?
Can Photomath do factoring?

Factor expressions step by step

The calculator will try to factor any expression (polynomial, binomial, trinomial, quadratic, rational, irrational, exponential, trigonometric, or a mix of them), with steps shown. To do this, some substitutions are first applied to convert the expression into a polynomial, and then the following techniques are used: factoring monomials (common factor), factoring quadratics, grouping and regrouping, square of sum/difference, cube of sum/difference, difference of squares, sum/difference of cubes, and the rational zeros theorem.

Solution

Your input: factor $$$x^{4} - 20 x^{2} + 64$$$.

We can treat $$$x^{4} - 20 x^{2} + 64$$$ as a quadratic function with respect to $$$x^{2}$$$.

Let $$$Y = x^{2}$$$.

Temporarily rewrite $$$x^{4} - 20 x^{2} + 64$$$ in terms of $$$Y$$$: $$$x^{4} - 20 x^{2} + 64$$$ becomes $$$Y^{2} - 20 Y + 64$$$.

To factor the quadratic function $$$Y^{2} - 20 Y + 64$$$, we should solve the corresponding quadratic equation $$$Y^{2} - 20 Y + 64=0$$$.

Indeed, if $$$Y_1$$$ and $$$Y_2$$$ are the roots of the quadratic equation $$$aY^2+bY+c=0$$$, then $$$aY^2+bY+c=a(Y-Y_1)(Y-Y_2)$$$.

Solve the quadratic equation $$$Y^{2} - 20 Y + 64=0$$$.

The roots are $$$Y_{1} = 16$$$, $$$Y_{2} = 4$$$ (use the quadratic equation calculator to see the steps).

Therefore, $$$Y^{2} - 20 Y + 64 = \left(Y - 16\right) \left(Y - 4\right)$$$.

Recall that $$$Y = x^{2}$$$: $$$x^{4} - 20 x^{2} + 64 = 1 \left(x^{2} - 16\right) \left(x^{2} - 4\right)$$$.

$$\color{red}{\left(x^{4} - 20 x^{2} + 64\right)} = \color{red}{1 \left(x^{2} - 16\right) \left(x^{2} - 4\right)}$$

Apply the difference of squares formula $$$\alpha^{2} - \beta^{2} = \left(\alpha - \beta\right) \left(\alpha + \beta\right)$$$ with $$$\alpha = x$$$ and $$$\beta = 2$$$:

$$\left(x^{2} - 16\right) \color{red}{\left(x^{2} - 4\right)} = \left(x^{2} - 16\right) \color{red}{\left(x - 2\right) \left(x + 2\right)}$$

Apply the difference of squares formula $$$\alpha^{2} - \beta^{2} = \left(\alpha - \beta\right) \left(\alpha + \beta\right)$$$ with $$$\alpha = x$$$ and $$$\beta = 4$$$:

$$\left(x - 2\right) \left(x + 2\right) \color{red}{\left(x^{2} - 16\right)} = \left(x - 2\right) \left(x + 2\right) \color{red}{\left(x - 4\right) \left(x + 4\right)}$$

Thus, $$$x^{4} - 20 x^{2} + 64=\left(x - 4\right) \left(x - 2\right) \left(x + 2\right) \left(x + 4\right)$$$.

Answer: $$$x^{4} - 20 x^{2} + 64=\left(x - 4\right) \left(x - 2\right) \left(x + 2\right) \left(x + 4\right)$$$.

My former algebra tutor got impatient whenever I couldn't figure out an equation. I eventually got tired of her so I decided to try the software. I'm so impressed with it! I can't stress enough how great it is!
Zoraya Christiansen

A solid software and we need more like it. Good job.
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Miguel San Miguel-Gonzalez, Laredo Int. University

What you will learn in this mini lesson

You will learn how to factor and solve Quadratic Equations in Standard Form when the leading coefficient A=1 and also when A ≠ 1. Our Step by Step Calculator allows you to factor and solve your own quadratic equation.

Quick Example when A ≠ 1
2x2 -12x+16 Factor out 2:
= 2(x2 -6x+8)
= 2(x-4)*(x-2) since (-4) + (-2) = -6 and (-4)*(-2)=8
Solving (x-4)=0 and (x-2)=0 yields the two zeros
x=4 and x=2 . That’s all 😉

How do I Factor Quadratic Equations?

A Quadratic Function in Standard Form :
x2+bx+c

In Factored Form it looks like this:
(x+r)*(x+s) where r,s are the 2 Zeros.

When distributing we get:
x2+(r + s)*x + r*s

Matching the Coefficients on both sides
x2+bx+c = x2 + (r + s)*x + r*s

shows that the 2 Zeros r and s have to fulfill the 2 conditions:
1) r+s = b and
2) r*s = c

In Words:
1) r and s have to add to the value of the middle coefficient b.
2) r and s multiplied have to equal the constant coefficient c.

What if the leading coefficient A is not 1 ?

Let’s factor Ax²+Bx+C=0 with A ≠ 1 .
We first divide the entire equation by A to get:
x²+(B/A)x+C/A = 0

Setting b=B/A and c=C/A we rewrite as
x2+bx+c=0

The Factored Form looks like this:
(x+r)*(x+s) = 0 – r,s are the 2 Zeros.

Distributing terms we get
(x2+(r + s)*x + r*s) = 0

We again Match the Coefficients:
x2+bx+c = x2 + (r + s)*x + r*s

It shows that the 2 Zeros r and s have to fulfill these 2 conditions:
1) r+s = b = B/A and
2) r s = c = C/A

In Words:
The 2 zeros r and s have to add to b = B/A.
And when multiplied equal c = C/A.
See below’s examples.

Sample Problem: How to Factor a Quadratic Equation?

1) Factor Quadratic Equations with Leading coefficient A = 1
We are to factor the Quadratic Equation
x2– 6x+8 .
The 2 zeros when multiplied have to equal 8.
That could be 8 and 1 OR 4 and 2, and their negatives.

Additionally, they have to add to -6 which implies
the 2 zeros must be -4 and -2.

Therefore, the factored version is:
x2– 6x+8 = (x-4)*(x-2) .

When asked to solve the Quadratic Equation
x2– 6x+8=0 .

we use the above factored version and set each factor equal to 0:
Since x-4=0 we get x=4 ,
and since x-2=0 we get x=2 .

Thus, the 2 zeros are x=4 , x=2

Easy, wasn’t it?

Tip: When using the above Factor Quadratic Equation Solver to factor
x2-6x+8 we must enter the 3 coefficients as
a=1, b=-6 and c=8.

2) Factor Quadratic Equations when A ≠ 1
We are to factor the Quadratic Equation
2x2– 12x+16 .
First divide by 2 to have a leading coefficient coefficient of A=1.
We get x2– 6x+8 as we had in the above example.

Since
x2– 6x+8 = (x-4)*(x-2)

we multiply by A=2 to get
2x2– 12x+16 = 2*(x-4)*(x-2)

as the factored form.

Tip: When using the above Factor Quadratic Equation Solver to factor
2x2-12x+16
we must enter the 3 coefficients a,b,c as
a=2, b=-12 and c=16.

This Video gives a great explanation on how to factor quadratic equations when the leading coefficient is not 1:

How do you solve a quadratic equation in factored form?

To solve an quadratic equation using factoring :.

1 . Transform the equation using standard form in which one side is zero..

2 . Factor the non-zero side..

3 . Set each factor to zero (Remember: a product of factors is zero if and only if one or more of the factors is zero)..

4 . Solve each resulting equation..

Can Photomath do factoring?

Just because math is getting more complicated doesn't mean we can't make it a little easier for ourselves! Factoring is one way we can make our calculations a little clearer and easier to follow.