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Basics of Quadratic Equations | Mathematics (Maths) Class 11 - Commerce PDF Download

What are quadratic equations?

A quadratic equation is an equation with a variable to the second power as its highest power term. For example, in the quadratic equation 3x2 - 5x-2=0:

  • x is the variable, which represents a number whose value we don't know yet.
  • The 2 is the power or exponent. An exponent of 2 means the variable is multiplied by itself.
  • 3 and -5 are the coefficients, or constant multiples of x2 and x. 3x2 is a single term, as is -5x.
  • -2 is a constant term.

How do I solve quadratic equations using square roots?

When can I solve by taking square roots?

Quadratic equations without x-terms such as 2x2 = 32 can be solved without setting a quadratic expression equal to 0. Instead, we can isolate x2 and use the square root operation to solve for x.
When solving quadratic equations by taking square roots, both the positive and negative square roots are solutions to the equation. This is because when we square a solution, the result is always positive.
For example, for the equation x2 = 4, both 2 and -2 are solutions:

  • 22 = 4
  • (-2)2 = 4

When solving quadratic equations without x-terms:

  • Isolate x2.
  • Take the square root of both sides of the equation. Both the positive and negative square roots are solutions.

Example: What values of x satisfy the equation 2x2 = 18?
Sol: 
Basics of Quadratic Equations | Mathematics (Maths) Class 11 - Commerce
The following values of x satisfy the equation 2x2 = 18:

  • -3 and 3

Question for Basics of Quadratic Equations
Try yourself:
What is the solution to the quadratic equation x^2 - 9 = 0?
View Solution

Zero Product Property and Factored Quadratic Equations

The zero product property states that if ab = 0, then either a or b is equal to 0.
The zero product property lets us solve factored quadratic equations by solving two linear equations. For a quadratic equation such as (x-5)(x + 2) = 0, we know that either x -5 = 0 or x + 2 = 0. Solving these two linear equations gives us the two solutions to the quadratic equation.
Basics of Quadratic Equations | Mathematics (Maths) Class 11 - Commerce
To solve a factored quadratic equation using the zero product property:

  • Set each factor equal to 0.
  • Solve the equations from Step 1. The solutions to the linear equations are also solutions to the quadratic equation.

Solving Factorable Quadratic Equations

If we can write a quadratic expression as the product of two linear expressions (factors), then we can use those linear expressions to calculate the solutions to the quadratic equation.
We'll focus on factorable quadratic equations with 1 as the coefficient of the x2 term, such as x-2x - 3 = 0. For more advanced factoring techniques, including special factoring and factoring quadratic expressions with x2 coefficients other than 1, check out the Factoring quadratic and polynomial expressions.
Recognizing factors of quadratic expressions takes practice. The factors will be in the form (x + a) (x + b), where a and b fulfill the following criteria:

  • The sum of a and b is equal to the coefficient of the x-term in the unfactored quadratic expression.
  • The product of a and b is equal to the constant term of the unfactored quadratic expression.

For example, we can solve the equation x2 - 2x - 30 by factoring x2 - 2x - 3 into (x + a) (x + b), where:

  • a + b is equal to the coefficient of the x-term, -2.
  • ab is equal to the constant term, -3.

-3 and 1 would work:

  • -3+1-2
  • (-3)(1) = -3

This means we can rewrite x2 - 2x - 3 = 0as (x - 3)(x + 1) = 0 and solve the quadratic equation using the zero product property. Keep mind that a and b are not themselves solutions to the quadratic equation!
Basics of Quadratic Equations | Mathematics (Maths) Class 11 - Commerce

When solving factorable quadratic equations in the form x2 + bx + c = 0:

  • Rewrite the quadratic expression as the product of two factors. The two factors are linear expressions with an x-term and a constant term. The sum of the constant terms is equal to b, and the product of the constant terms is equal to c.
  • Set each factor equal to 0.
  • Solve the equations from Step 2. The solutions to the linear equations are also solutions to the quadratic equation.

The Quadratic formula

Not all quadratic expressions are factorable, and not all factorable quadratic expressions are easy to factor. The quadratic formula gives us a way to solve any quadratic equation as long as we can plug the correct values into the formula and evaluate.
Basics of Quadratic Equations | Mathematics (Maths) Class 11 - Commerce

Question for Basics of Quadratic Equations
Try yourself:
What does the zero product property state?
View Solution

What are the steps?
To solve a quadratic equation using the quadratic formula:

  • Rewrite the equation in the form ax2 + bx + c = 0.
  • Substitute the values of a, b, and c into the quadratic formula, shown below.
    Basics of Quadratic Equations | Mathematics (Maths) Class 11 - Commerce
  • Evaluate x.
The document Basics of Quadratic Equations | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Basics of Quadratic Equations - Mathematics (Maths) Class 11 - Commerce

1. What are quadratic equations and why are they important in mathematics?
Ans. Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They are important in mathematics as they are used to solve a variety of real-world problems and have applications in fields such as physics, engineering, and economics.
2. How can square roots be used to solve quadratic equations?
Ans. To solve a quadratic equation using square roots, you can isolate the squared term on one side of the equation and then take the square root of both sides. This will help you find the values of x that satisfy the equation.
3. What is the Zero Product Property and how is it used to solve quadratic equations?
Ans. The Zero Product Property states that if the product of two or more factors is equal to zero, then at least one of the factors must be zero. This property is used to solve quadratic equations by setting the equation equal to zero and then factoring it to find the values of x that make the equation true.
4. How do you solve factorable quadratic equations?
Ans. To solve factorable quadratic equations, you can factor the equation into two binomials and set each binomial equal to zero. This will help you find the values of x that satisfy the equation.
5. What is the Quadratic formula and how is it used to solve quadratic equations?
Ans. The Quadratic formula is used to find the roots of a quadratic equation of the form ax^2 + bx + c = 0. It is given by x = (-b ± √(b^2 - 4ac)) / 2a. By substituting the values of a, b, and c into the formula, you can find the values of x that satisfy the quadratic equation.
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