Algebra

Quadratic Form Calculator

Published by
Mike Woodhill PhD

What you will learn in this mini lesson

You will learn how to solve Quadratic Forms with our Calculator using the famous Quadratic Formula. Our Step by Step Calculator allows you to solve your own quadratic equation.


How do I Solve Quadratic Forms?

Any Quadratic Equation can be written in the Standard Form:

When finding the zeros its 2 solutions can be found by applying the famous quadratic formula:

Example: Solve a Quadratic Form using The Quadratic Formula

To solve 2x²+5x-3=0 using the Quadratic Formula we enter the coefficients a=2, b=5 and c=-3 into the above formula to get

Simplification yields

Since 25+24=49 we have

The Square Root of 49 is known to be 7

One solution is found by adding 7 , the second solution by subtracting 7

Simplifying both fractions finally yields the 2 real solutions

These two solutions can be verified by plugging them back into the given above Quadratic Equation.

For visual learners please watch the video below on how to Solve a Quadratic Form:


You may use our Quadratic Form Calculator below to solve your own exercises.


The two ways to solve a Quadratic Forms in Vertex Form

We can solve a Quadratic Forms in Vertex Form y = a(x-h)²+k in 2 ways.
(a) We transform it into Standard Form to use the above Quadratic Formula again.
Example: 2(x+5/4)²-49/8=0 .
Expanding yields: 2(x²+5/4x+25/16)-49/8=0
Distributing 2 yields:
2x²+5x+25/8-49/8=0
Since 25/8 – 49/8 = -24/8 = -3 :
We finally have 2x²+5x-3=0 .
We again use the above Quadratic Formula to arrive at the 2 solutions x=1/2 and x=-3.

(b) Alternatively, we simply use Algebra and square roots to find the 2 solutions to 2(x+5/4)²-49/8=0 . Here is how:
Add 49/8: 2(x+5/4)²=49/8 .
Divide by 2: (x+5/4)² = 49/16 .
Square Roots: x+5/4 = ±7/4 .
Thus, x = -5/4 – 7/4 = -2/4 and x= -5/4 + 7/4 = 2/4 .
The 2 solutions are of course x=1/2
and x=-3

Exercise for you

Solve y=(x-2)²-9 = 0 in 2 ways:
First, convert to Standard form next use Algebra.

DON’T CONTINUE READING UNTIL DONE.


We first apply the binomial formula to expand and get
(x²-4x+4)-9=0
With 4-9=-5 we finally arrive at the Standard Form:
y=x²-4x-5=0
Solving using the Quadratic Formula yields: x=5 , x=-1

We created a separate page for you that teaches the Vertex to Standard Form Conversion. It also has a Solver that allows you convert you Vertex Form Equation into Standard Form. Visit our page here.
Below, you may use our Vertex to Standard Form Converter.


Finally, learn the Quadratic Form Song to never forget the formula again. Pop goes the weasel. Have fun 😉
Mike Woodhill PhD

As a long time instructor I would like you to benefit from my Step by Step Calculators. You may also schedule a live tutoring session to succeed in your class. 😉

Published by
Mike Woodhill PhD

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