# Why Do We Solve Quadratic Equations?

Students often wonder why mathematicians solve quadratic equations. Some have accused them of subjecting students to quadratic equations as cruel torture.

There are various approaches to solving quadratic equations, but three methods that stand out among them are factoring, using the quadratic formula and completing the square.

## Solving Quadratic Equations

Quadratic equations are a mainstay in mathematics, often appearing in word problems such as finding the perimeter of a square, solving for speed or calculating area of a rectangle. There are three methods for solving quadratic equations: factoring, using the quadratic formula and completing the square.

Step one in solving a quadratic equation is converting the given equation to standard form: ax2 + bx3 + c4 = 0. Next comes looking for values for a, b and c; to find this information the discriminant must be calculated; it serves to show whether solutions to quadratic equations will be real numbers, complex numbers, irrational numbers or multiples thereof. If this value equals 0, there will no complex or irrational solutions and it should be easy to solve.

If the discriminant is negative, identifying correct values for a, b and c may be difficult; however, if the equation is positive it will be easier to find out what a means; from there it can be determined which other values to set for each variable and plug them into the quadratic formula to solve for x.

Once a solution has been identified, it’s important to evaluate its validity by comparing it against the original problem statement. If they match up perfectly then we have found our answer; otherwise we must find alternative means of solving this puzzle.

There are various methods available to solve quadratic equations, but the easiest and most reliable one is through using the quadratic formula. Not only can its values of a, b and the constant be derived easily from any given equation, but this method of solution can also work for any quadratic equation that exists in standard form as well as any value for either of its variables a, b and/or c.

## Finding the Roots

There are countless real-life situations we can model with quadratic equations, and one key aspect we need to keep in mind when solving them is what happens when both y-coordinates reach zero (known as the root). Knowing this allows us to predict what will happen when something such as launching a projectile, running out of catalyst in chemical reaction or skidding occurs among other events.

Factoring quadratic equations is the simplest and most basic method of finding its roots, and will likely be taught as part of any quadratic algebra course. When factoring a quadratic equation we start by collecting all x2 terms on one side and all constants on the other before taking square roots of both sides to identify values of x that will produce factors on its right-hand side of the equation.

An alternative method for quickly finding roots of a quadratic equation faster than factoring is using the Quadratic Formula. This shorthand can be applied when using standard form quadratic equations to identify values for a, b and c.

When we use the Quadratic Formula to solve an equation, we typically get two real roots if its discriminant is positive, and one if its discriminant equals 0. If a quadratic equation lacks real roots altogether, however, then its solution cannot be determined using this method.

Completing the Square (Completing the Square is a method used to find quadratic roots that works both with real and imaginary numbers), starting by writing out our quadratic equation in this format: (ax2 + bx + c = 0); this should ensure that a = 1.

From there, we take the value of b from this new equation and divide both sides by it to get the value of a. Once that value has been obtained, it can be added back into both sides to form a perfect square. Repeating this process for any other values such as c yields a and b values necessary for solving quadratic equations.

## Finding the Square Roots

Roots of a quadratic equation are found at points where its graph intersects the x-axis, while solutions consist of any numbers which satisfy its equation, either real or complex depending on its discriminant value; positive values will have two real solutions while zero will result in just one solution, while negative discriminant values will result in no solution at all.

To solve a quadratic equation, calculate the square root of each side and add them together before solving for x. Finally, simplify your answer.

This method of solving quadratic equations is known as completing the square and is taught in most college algebra courses. There are other approaches for solving quadratic equations as well, including factoring. No matter what approach is chosen to find solutions, it is imperative that any solutions found are simplified for simplicity’s sake.

One alternative method of finding the roots of a quadratic equation is using the quadratic formula. Though less popular than completing the square method, this approach can prove invaluable in certain circumstances. This shortcut method works when both coefficients of the x2 term and x term are equal – for instance when writing out your equation in standard form then substituting these values into the quadratic formula. To use it effectively you will need a set of three values identified as A, B, and C into your standard form equation and into its quadratic formula as A B and C into its final form before substituting into its final form to use this method effectively.

Once we know the values for a, b and c it is easy to calculate how many real or complex solutions there will be – this number depends on the value of the discriminant of the quadratic equation which can be calculated by multiplying their powers together.

Solution of a quadratic equation can also be obtained by analyzing its graph. A convex graph will have two real solutions, while concave graphs may only contain one solution while non-convex ones might result in no solution at all.

## Finding the Intercepts

Quadratic equations can be used to explain an array of mathematical phenomena, such as population change over time or how much water it takes to fill a cup. Finding both the x- and y-intercepts are crucial parts of quadratic functions and require an ability to locate them; once learned this skill can also help determine slope or arch height problems.

The y-intercept is the point at which a line intersects with the y-axis and can be determined in various ways, depending on your available information. If the line is written in slope-intercept form (y=mx+b), algebra will make this task easy; otherwise you could determine it from graphing or an equation such as “y=-3×2 – 3x + 1”, where when graphed its value will correspond with when x = 0. You could also solve for its x-intercept using its value to solve for its value when solving for its value when solving for its value of “y”.

One method of finding the y-intercept is calculating its discriminant value using either an online calculator or pen and paper. This value reveals the nature of quadratic equation roots; you can do this by dividing square root of quadratic formula by 2, and finding its value. If it equals 0, that indicates real and equal roots while negative numbers indicate imaginary or undetermined roots.

Know How to Solve Quadratic Equations by Hand [Hand] If you know how to solve quadratic equations manually, this knowledge can quickly assist in finding the y-intercept of any function or equation without an x-intercept; for instance y=3×2 – 2x+3 can be solved by writing out this formula y=0x2 + bx+c and solving for y. Doing this reveals that its y-intercept is always constant at c when looking at its graph; which will show its crossing with the y-axis at some constant value on its graph.