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## Algebra 1

Gain more insight into the quadratic formula and how it is used in quadratic equations.
The quadratic formula helps you solve quadratic equations, and is probably one of the top five formulas in math.  We’re not big fans of you memorizing formulas, but this one is useful (and we think you should learn how to derive it as well as use it, but that’s for the second video!).
If you have a general quadratic equation like this:
a, x, squared, plus, b, x, plus, c, equals, 0
Then the formula will help you find the roots of a quadratic equation, i.e. the values of x where this equation is solved.

x, equals, start fraction, minus, b, plus minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction
It may look a little scary, but you’ll get used to it quickly!
Practice using the formula now.

### Worked example

First we need to identify the values for a, b, and c (the coefficients). First step, make sure the equation is in the format from above, a, x, squared, plus, b, x, plus, c, equals, 0:
x, squared, plus, 4, x, minus, 21, equals, 0
• a is the coefficient in front of x, squared, so here a, equals, 1 (note that a can’t equal 0 -- the x, squared is what makes it a quadratic).
• b is the coefficient in front of the x, so here b, equals, 4.
• c is the constant, or the term without any x next to it, so here c, equals, minus, 21.
Then we plug a, b, and c into the formula:
x, equals, start fraction, minus, 4, plus minus, square root of, 16, minus, 4, dot, 1, dot, left parenthesis, minus, 21, right parenthesis, end square root, divided by, 2, end fraction
solving this looks like:
\begin{aligned} x&=\dfrac{-4\pm\sqrt{100}}{2} \\\\ &=\dfrac{-4\pm 10}{2} \\\\ &=-2\pm 5 \end{aligned}
Therefore x, equals, 3 or x, equals, minus, 7.

## What does the solution tell us?

The two solutions are the x-intercepts of the equation, i.e. where the curve crosses the x-axis. The equation x, squared, plus, 3, x, minus, 4, equals, 0 looks like:
where the solutions to the quadratic formula, and the intercepts are x, equals, minus, 4 and x, equals, 1.
Now you can also solve a quadratic equation through factoring, completing the square, or graphing, so why do we need the formula?
Because sometimes quadratic equations are a lot harder to solve than that first example.

## Second worked example

Let’s try this for an equation that is hard to factor:
3, x, squared, plus, 6, x, equals, minus, 10
Let’s first get it into the form where all terms are on the left-hand side:
start underbrace, left parenthesis, 3, right parenthesis, end underbrace, start subscript, a, end subscript, x, squared, plus, start underbrace, left parenthesis, 6, right parenthesis, end underbrace, start subscript, b, end subscript, x, plus, start underbrace, left parenthesis, 10, right parenthesis, end underbrace, start subscript, c, end subscript, equals, 0
The formula gives us:
\begin{aligned} x&=\dfrac{-6\pm\sqrt{6^2-4\cdot 3\cdot 10}}{2\cdot 3} \\\\ &=\dfrac{-6\pm\sqrt{36-120}}{6} \\\\ &=\dfrac{-6\pm\sqrt{-84}}{6} \end{aligned}
We know you can’t take the square root of a negative number without using imaginary numbers, so that tells us there’s no real solutions to this equation.  This means that at no point will y, equals, 0, the function won’t intercept the x-axis.  We can also see this when graphed on a calculator:
Quadratic equation shown on a calculator
Now you’ve got the basics of the quadratic formula!
There are many more worked examples in the videos to follow.

## Tips when using the quadratic formula

• Be careful that the equation is arranged in the right form: a, x, squared, plus, b, x, plus, c, equals, 0 or it won’t work!
• Make sure you take the square root of the whole left parenthesis, b, squared, minus, 4, a, c, right parenthesis, and that 2, a is the denominator of everything above it
• Watch your negatives: b, squared can’t be negative, so if b starts as negative, make sure it changes to a positive since the square of a negative or a positive is a positive
• Keep the plus, slash, minus and always be on the look out for TWO solutions
• If you use a calculator, the answer might be rounded to a certain number of decimal places. If asked for the exact answer (as usually happens) and the square roots can’t be easily simplified, keep the square roots in the answer, e.g. start fraction, 2, minus, square root of, 10, end square root, divided by, 2, end fraction and start fraction, 2, plus, square root of, 10, end square root, divided by, 2, end fraction

## Next step:

• Watch Sal do an example:
Using the quadratic formulaSee video transcript
Proof of the quadratic formulaSee video transcript

## Want to join the conversation?

• does x2 = x to the power of 2?
• Yes x with a little 2 to its top right is x to the power of 2, but for future reference when typing x to the power of 2 on the computer the convention is to use the "^" symbol to say "to the power of"

so x to the power of 2 would be x^2
• Just curious, is there something like the "Trinomial formula", for third degree polynomials and so on? Or do we figure it out by normal factorization? So what makes second degree polynomials so special over say, 5th, or 3rd degree ones?
• Good question!
First note, a "trinomial" is not necessarily a third degree polynomial. A trinomial is a polynomial with 3 terms. It can have any degree. A third degree polynomial is called a cubic polynomial. Similar to how a second degree polynomial is called a quadratic polynomial.
There are general formulas for 3rd degree and 4th degree polynomials as well. These are the cubic and quartic formulas. Both of these formulas are significantly more complicated and difficult to derive than the 2nd degree quadratic formula! Here is a picture of the full quartic formula:
Be sure to scroll down and to the right to see the full formula! It's huge! In practice, there are other more efficient methods that we can employ to solve cubics and quartics that are simpler than plugging in the coefficients into the general formulae.
In fact, the highest degree polynomial that we can find a general formula for is 4 (the quartic). The Abel-Ruffini Theorem establishes that no general formula exists for polynomials of degree 5 or higher. So it's not that we haven't yet found a formula for a degree 5 or higher polynomial. It's that we will never find such formulae because they simply don't exist. You can read about the theorem here:
https://en.wikipedia.org/wiki/Abel–Ruffini_theorem
So in conclusion, there are only general formulae for 1st, 2nd, 3rd, and 4th degree polynomials. No such general formulas exist for higher degrees.
• How do i know when the curve goes like a u or a upside down u ?
• If the coefficient of x^2 is negative, the curve will look like an upside down u (i.e. the curve will have an absolute maximum). If the coefficient of x^2 is positive, the curve will look like a u (i.e. the curve will have an absolute minimum).

Hope this helps.
• Is anyone here in 2022
• instead of the formula, my textbook wants me to use factorization..how to i do x^2+2x-3=0?
1. how do i factorize x^2+2-3?
2. is it possible to use the formula for this? (i tried but cldnt seem to find the answer
(1 vote)
• if you mean find the solution, yes, you would get -3 and 1.
If you want to factor it would be (x + 3) (x - 1).
The quadratic formula helps you find the roots not the factored form.
• Could you extend this quadratic formula to work for other non-linear equations as well? I mean I have heard of so called Octic Equations which are of the form:

ax^8 + bx^7 + cx^6 + dx^5 + ex^4 + fx^3 + gx^2 + hx + i

and no I am not using d to mean derivative, or e to mean 2.7... or f, g, and h to mean function of x or i to mean the imaginary unit, just as variables.
• In 1827, a mathematician by the last name of Abel proved that there is no way to make an analogous equation past the 4th degree. One example (I found all of this on the cubic equation link) is the inverse of the function f(x)=x^5+x. There is simply no way to make an analogous equation for any polynomial of degree y for y>4, not enough operations are defined by the rules of mathematics. Maybe someone who reads this could invent one? : )