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Quadratic equations are an important part of algebra, and as students, we must all be familiar with their definition and the ways of solving quadratic equation problems. In this article, we are going to familiarize the students with all the concepts surrounding quadratic equations and the methods of solving problems related to this topic. A short definition of a quadratic equation would be: a quadratic equation is a second-degree polynomial, which we represent as axÂ² + bx + c in general.

In this representation, a cannot be equal to 0 and b,c are known as coefficients and are constant by nature. With this basic introduction, let's move forward with a formal definition, formulae and detailed solutions to quadratic equation questions to enable better understanding.

A quadratic equation is a polynomial where the highest power of the variable is 2. We generally represent it as axÂ² + bx + c. Here a, b and c are real numbers or constants, and x is the variable. In this case, the value of a cannot be 0 as that would remove the xÂ² term, and the equation won't be quadratic after that.

Some examples of quadratic equations can be as follows:

56xÂ² + â…” x + 1, where a = 56, b = â…” and c = 1.

-4/3 xÂ² + 64x - 30, where a = -4/3, b = 64 and c = -30.

To solve basic quadratic equation questions or any quadratic equation problems, we need to solve the equation. Solving quadratic equations gives us the roots of the polynomial. The roots of the equation are the values of x at which axÂ² + bx + c = 0. Since a quadratic equation is a polynomial of degree 2, we obtain two roots in this case.

There are several methods for solving quadratic equation problems, as we can see below:

Factorization Method.

Completing The Square Method.

Quadratic Equation Formula.

So what is the quadratic equation formula? The quadratic equation formula or the Sridharacharya Formula is a method for finding out the roots of two-degree polynomials. This formula helps solve quadratic equation problems. The formula is as given below:

x= {-b +/- (bÂ²-4ac)Â¹^{/}Â² }/2a

[where x represents the roots of the equation and bÂ² - 4ac is the discriminant].

By finding out the value of the discriminant, we can predict the nature of the roots. There are three possibilities with three different implications:

Two distinct roots which are real, if bÂ² - 4ac > 0.

Two real roots equal in magnitude, if bÂ² - 4ac = 0.

Imaginary roots or absence of real roots if bÂ² - 4ac < 0.

Now that the basic principles of quadratic equations are clear, we will move onto some solved examples. But before that, let us list some quadratic equation questions for the students to solve.

The following are a list of questions for you to solve once you have gone through the quadratic equation questions and answers in the solved examples section:

Find the determinant of the following quadratic equations: 2xÂ² + 3x + 6, 70xÂ² + 49 + 14, â…” yÂ² + 63y + 42.

Find the roots of the following quadratic equations: xÂ² - 45x + 324, 2xÂ² - 22x + 42, Â½ xÂ² + 2x + 4.

The product of two consecutive numbers is 420, and their sum is 41. Find the numbers.

Before solving these, let's check out the solved examples with questions and answers on the quadratic equation.

Q1. Solve xÂ² + 5x + 6 = 20 by factorization method.

Answer: The given polynomial or quadratic equation is

xÂ² + 5x + 6 = 20

Solving by factorization method,Â

xÂ² + 5x + 6 - 20 = 0.

or, xÂ² + 5x - 14 = 0

or, xÂ² - 2x + 7x - 14 = 0

or, x(x - 2) +7(x - 2) = 0

or, (x - 2)(x + 7) = 0

or, (x - 2) = 0, (x + 7) = 0Â

or, x = +2, -7.

Q2. Solve 2xÂ² - 5x + 3 using the quadratic equation formula.

Answer: The quadratic equation formula is:

x= {-b +/- (bÂ²-4ac)Â¹^{/}Â² }/2a

The determinant or bÂ²-4ac = (-5)Â² - 4 Ã— 3 Ã— 2 = 25 - 24 = 1

or, (bÂ² - 4ac)Â¹^{/}Â² = 1

Therefore, x = { -(-5) + 1}/2 Ã— 2= 6/4 = 3/2Â

or, x = {-(-5) - 1}/2 Ã— 2 = 4/4 = 1

Thus the roots of the equation are 3/2 and 1.

FAQ (Frequently Asked Questions)

Q1. Define a Quadratic Equation Along with Suitable Examples. Also, State the Quadratic Equation Formula.

Answer: A quadratic equation is a polynomial where the highest power of the variable is neither more nor less than 2. So essentially, a quadratic equation is a polynomial of degree 2. We represent such an equation in a general format as axÂ² + bx + c, where a, b and c are known as the coefficients or the constants of the equation. The thumb rule for quadratic equations is that the value of a cannot be 0. The x in the expression is the variable. This algebraic expression, when solved, will yield two roots.

Some examples of quadratic equations are:

3xÂ² + 4x + 7 = 34

xÂ² + 8x + 12 = 40

The quadratic equation formula is a method for solving quadratic equation questions. The formula is as follows:

x= {-b +/- (bÂ²-4ac)Â¹^{/}Â² }/2a

where x represents the roots of the equation.

Q2. What are the Roots of a Quadratic Equation? What are the Zeroes of a Polynomial? What are the Methods of Finding the Roots?

Answer: The roots of a quadratic equation are the values obtained when we solve the equation. They are those values of x for which the expression axÂ²+bx+c becomes equal to 0. These values are also known as the zeroes of the polynomial. Since a quadratic equation is essentially a polynomial of degree 2, we get two roots after solving the given polynomial.

To solve quadratic equation problems, we have several methods to help us out. We are listing these below:

The Factorization Method.

Completing The Square Method.

The Quadratic Equation Formula Method.

As we practice more and more quadratic equation sums, our ideas regarding which method to use while solving a given question will get clearer.