The points where this curve crosses the x axis are represented by the second form of the equation:
These are solved for using the quadratic formula, which will not only solve for real roots, but result in the imaginary roots if the parabola does not actually cross the x axis (this is when is greater than ).
Quadratic equations have been known about for thousands of years. The earliest known solution to a quadratic equation is often credited to the Babylonians from Berlin papyrus from the Middle Kingdom in Egypt and dated to around 2160-1700 BC. However, the Babylonians did not have the concept of an equation and they only considered positive solutions as solutions normally referred to a length. The ancient Greeks are known to have solved quadratic equations using geometry. Euclid described three problems described by quadratic equations in his Data and developed such a geometric approach in around 300 BC. Like with the Babylonians, it too only admitted positive solutions.
Several Indian mathematicians advanced ideas developed by the Babylonians. In the seventh century, a mathematician named Brahmagupta produced an almost modern method that allows for negative solutions. Brahmagupta also recognized that quadratics always have two solutions. Indian mathematics also developed the concept of negative numbers and zero which allowed a more complete understanding of such equations. The works of other Indian mathematicians at this time suggest they may have had an understanding of quadratic equations, though didn't explicitly describe them.
A century later in 820&nsbp;AD, the Arabic mathematician Al-Khwarismi developed an understanding of quadratic equations and a formula for solving them but rejected negative solutions.
Later in 1545, Girolamo Cardano compiled all that was known about quadratic equations at the time into a single work. He suggested the existence of imaginary numbers and complex numbers. In the late 16th century, François Viète introduced the modern mathematical notation used for quadratics.
Solving quadratic equations
There are several methods that can be used to solve quadratic equations. Different methods are easier to use depending on the equation one wishes to solve. Below are the main methods used.
Quadratic equations can always be written as the product of two linear factors of x:
When the equation is written in this form, it is said to be "factorized". Since we want to solve the quadratic equation being equal to zero, we can say that,
which means that either (x+r) or (x+s) are equal to zero. This means that -r and -s are the solutions to the equation. To see how we can factorise the quadratic it's simplest to consider the case of a=1 first,
Comparing the coefficients of the x and constant terms, r and s should add together to equal b and multiply to give c. To solve a quadratic equation by factorising, you should therefore try to guess a pair of numbers that meet these conditions. It is often easier to work methodically through pairs of factors of c. It is important to note that the solution will be the negative of these numbers.
To solve a quadratic equation when a is not 1, the a can simply be factored out,
and the term in brackets solved as above.
We want to choose r and s so they add to give 5 and multiply to give 6. Suppose we pick r=1 and s=6 to begin with since these are factors of 6. We then have r×s=1×6=6 so the multiplication condition is met. However 1+6=7 and not 5, so the addition condition is not satisfied; we have not found the solution.
Next suppose we try r=2 and s=3. Then we have r×s=2×3=6 and r+s=2+3=5. This satisfies both conditions so we can say the solution is (remembering the solutions are -r and -s) x=-2 and x=-3.
Completing the square
Completing the square is a particularly useful method that will always work, but takes longer and is more complicated than factorizing. The quadratic we want to solve is,
Again consider the case of a=1 for simplicity first. Completing the square works by first rewriting the equation into the form,
where α and β are two numbers. Multiplying the right hand side of the equation out we see,
We therefore have,
and the original quadratic can be rewritten as,
From this it is straightforward to solve and get both solutions. Taking the constant terms onto the right hand side,
Taking the square root and remembering that we must take plus or minus,
Then subtracting the b/2 from both sides gives the solution:
The case of a not equal to 1 can be solved by factoring it out and leads to the quadratic formula.
We first write it in completed square form as,
Taking the right two terms onto the right hand side produces,
Then square rooting and remembering the plus and minus sign,
Therefore the solutions are,
which is x=2 when we use the plus and x=1 when we choose the minus.
To use it simply substitute in for the different values of a, b and c. If the quadratic equation you want to solve has unusual numbers or even algebraic expressions for any of a, b or c, this is likely the fastest and easiest way to solve it.
Substituting for a, b and c into the quadratic formula we get,
Performing some simplifying this is just,
Therefore the solutions are:
Now consider the following quadratic,
Applying the quadratic formula gives the solutions to be,
The number in the square root is negative meaning the solution to the quadratic will be complex. Since,
the solution to the quadratic is,
The expression for the quadratic formula above is not in fact unique. If one divides the original quadratic by x2 to get,
This formula is sometimes called the "Citardauq formula" to distinguish it from the usual quadratic formula and is useful when a is small.
Like with many equations, quadratic equations can be solved numerically. However, since any quadratic equation can be solved analytically (non-numerically) using the quadratic formula which provides and exact expression for the solution, this is not often used.
The discriminant and nature of solutions
In general, the a, b and c coefficients can be complex numbers. The solutions are then also complex numbers. However, if the a, b and c coefficients are restricted to being real numbers then the possible solutions fall into three categories described by the discriminant. The discriminant is the part inside the square root of the quadratic formula, namely,
The value of the discriminant can either be greater than zero, equal to zero or less than zero:
- If it is greater than zero, the quadratic has two real solutions.
- If it is zero, the quadratic has only one real solution. The solution is sometimes called a repeated root.
- If it is less than zero, the square root evaluates to an imaginary number. There are then two complex solutions which are complex conjugate of each other.
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