Solving a Quadratic Equation: Simple Steps

Haknem School
This guide from Haknem School provides you with a simple, step-by-step method for solving quadratic equations. It offers a comprehensive approach from understanding the formula to verifying the answers.
a chequered maths student's maths notebook the school that is open at the point where the quadratic equation is solved

A quadratic equation is an equation of the form ax2 + bx + c = 0, where a, b, and c are coefficients, and x is the variable we want to find. The coefficient a cannot be zero, otherwise the equation will no longer be quadratic.

Step 1: Understanding the formula

To solve a quadratic equation, we use the so-called quadratic formula:

Solving quadratic equations: The quadratic equation formula with coefficients and variables

In this formula, x are the solutions of the quadratic equation, a is the coefficient of x2, b is the coefficient of x, and c is the constant term. The symbol ± means that there are generally two solutions: one with addition (+) and one with subtraction (−). The square root symbol ​covers the expression b2−4ac, which is called the discriminant. The solutions are real if the discriminant is non-negative, and complex if the discriminant is negative.

Solving quadratic equations: The formula for calculating the discriminant in a quadratic equation

this expression under the root is called the discriminant.

Read also: what roots a quadratic equation will have depending on whether the discriminant — is greater than zero, equal to zero, or less than zero:

Step 2: Substituting the Coefficients

Now that we know the formula, we can substitute the coefficients from our equation into it. Let’s say we have the equation x2  2x + 1 = 0. Here, a = 1, b = − 2, and c = 1.

Step 3: Calculation

Now we simply substitute our coefficients into the formula and calculate the value of x:

Solving quadratic equations: Finding the roots of a specific quadratic equation using the quadratic formula and discriminant

As you can see, we got two answers: x1,2​x1 = 1 и x2​ = 1. This means that our equation has one root, which is 1. This is called a “double degenerate” root.

Step 4: Verification

After we have found the roots of the equation, we can check them by substituting them back into the equation. If the equation turns into a correct numerical equality when substituting the roots, it means we have found the correct roots.

In our case, if we substitute x=1 into the equation x2 − 2x + 1 = 0, we get 12 − 2∗1 + 1= 0, which is a correct equality. So, x=1 is indeed a root of the equation.

That’s it! Now you know how to solve quadratic equations. Remember, practice is the key to success in mathematics. The more equations you solve, the better you’ll understand their structure and the easier it will be for you to find solutions. And remember, every time you solve a math problem, you’re training your brain, developing logical thinking, and learning to overcome challenges. Good luck with your math studies!

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