Quadratic equations are an important topic of algebra that everyone should learn in their early classes. A quadratic equation is a polynomial equation that has a degree of order 2. In standard form, it is represented as ax² + bx + c = 0 where a, b, and c are constants, and x represents the variable. This quadratic equation has importance in other subjects also such as physics, engineering, economics, and more other than mathematics. In this article we are going to learn about quadratic equations, how factoring is used to solve quadratic equations, and some practice questions based on it.

What is a Quadratic Equation?

Quadratic equations as the word says quad – means square, which indicates it is a second-degree polynomial equation. A quadratic equation is a second-degree polynomial equation in a single x. The standard form of writing quadratic equations is:

ax² + bx + c = 0

where:

• a is the coefficient x² (the quadratic term)
• b is the coefficient of x (the linear term)
• c is the constant term

Examples of Quadratic Equations

• 2x² + 3x − 5 = 0
• x² − 4x + 4 = 0
• 2x² + 6 = 0

Importance of Quadratic Equations to Students

Quadratic equations are used in various real-life situations, such as projectile motion, area calculations, and optimization problems. Mastery of solving quadratic equations is important for students pursuing science, technology, engineering, and mathematics.

Methods of Factoring Quadratic Equation

Some methods of factoring are mentioned below:

Factoring by Common Factors

The first step in factoring is to look for common factors in all terms of the quadratic equation. If a common factor is found, then it should be factored out.

Example:

2x² + 4x = 0

Both terms have a common factor of 2x:

2x (x + 2) = 0

Thus, the solutions of the above quadratic equation are x = 0 and x = -2

Factoring by Grouping

This method is used for solving when a quadratic equation can be split into groups that have common factors.

Example:

x² + 5x + 6 = 0

Rewrite the middle term (5x) as a sum of two terms whose coefficients add up to 5:

x² + 2x + 3x + 6 = 0

Group the terms of the equation:

(x² + 2x) + (3x + 6) = 0

Factor out the common factors from each group:

x(x + 2) + 3(x + 2) = 0

Factor out the common binomial factor:

(x + 2) (x + 3) = 0

Factoring Trinomials

When factoring trinomials of the form ax² + bx + c, we need to find two numbers that multiply to ac and add up to b.

Example:

x² + 7x + 12 = 0

The numbers 3 and 4 multiply by 12 and add up to 7:

(x + 3) (x + 4) = 0

Factoring Perfect Square Trinomials

A perfect square trinomial is a trinomial that can be represented as the square of a binomial.

Example:

x² + 6x + 9 = 0

This is a perfect square trinomial because:

(x+3)² = 0

Factoring by Completing the Square

This method of factoring is used when the quadratic equation cannot be factored easily using any traditional methods.

Steps Factoring by Completing Square

Step 1: Write quadratic equation in standard form: ax² + bx + c

Step 2: Move the constant term to another side.

Step 3: Now, take half of the coefficient of x, square it, and add it to both sides of the equation which creates a perfect square trinomial on one side.

Step 4: Now take square root of both side and solve for x.

Example: Solve x² + 6x + 5 = 0 by using the method of completing the square.

Solution:

Move the constant term to other side of equation:

x² + 6x = -5

Half the coefficient of x i.e. 6/2 = 3 , and 3² = 9

Add 9 to both side

x² + 6x + 9 = -5 + 9

(x + 3)² = 4

Take square root of both side:

(x + 3) = ± 2

x + 3 = +2 or x + 3 = -2

x = -1 or x = -5

Factoring the Difference of Squares

The difference of squares is a special factoring case where the quadratic equation is of the form a² −b².

Example: Factorize x² − 9 = 0

Solution:

x² − 9 = 0

This can be factored as:

(x + 3) (x − 3) = 0

Steps to Solve Quadratic Equations by Factoring

Follow the steps to solve Quadratic Equations by Factoring

Step 1: First write the quadratic equation in standard form: ax² + bx + c = 0. (where a ≠ 0)

Step 2: Use any one of the factoring methods discussed above to factor the quadratic equation.

Step 3: When factored, set each binomial factor equal to zero.

Step 4: Solve the resulting linear equations and find the value of x.

Example: Solve the given quadratic equation: x² – 4x + 4 = 0

Solution:

Equation is already in standard form ax² + bx + c = 0.

Factor the quadratic expression:

(x-2) (x-2) = 0

Now set each factor equal to zero:

x – 2 = 0 or x – 2 = 0

Now solve for x:

x = 2 or x = 2

Thus, the solutions of a quadratic equation are x = 2 and x = 2.

Examples on Quadratic equations by factoring

Example 1: Solve x² + 7x + 10 = 0

Solution:

Equation is in standard form

Factor the quadratic expression:

(x + 2) (x + 5) = 0.

Now set each factor equal to zero:

x + 2 = 0 or x + 5 = 0

Solve for

x = −2 or x = −5

Solution of above equation are x = -2 and x = -5

Example 2: Solve: x² – 5x + 6 = 0

Solution:

Equation is in standard form.

Factor the quadratic expression:

(x – 2) (x – 3) = 0.

Now set each factor equal to zero:

x – 2 = 0

x – 3 = 0

Solve for

x = 2 or x = 3

The solution of above equation are x = 2 and x = 3

Example 3: Solve: x² + 9x + 20 = 0

Solution:

Equation is in standard form.

Factor the quadratic expression:

(x + 4) (x + 5) = 0.

Now set each factor equal to zero:

x + 4 = 0

x + 5 = 0

Solve for

x = -4 or x = -5

The solution are x = -4 and x = -5

Example 4: Solve: x² – 3x – 10 = 0

Solution:

Equation is in standard form.

Factor the quadratic expression:

(x – 5) (x + 2) = 0.

Now set each factor equal to zero:

x – 5 = 0

x + 2 = 0

Solve for

x = 5 or x = -2

The solution are x = 5 and x = -2

Example 5: Solve: x² + 4x – 12 = 0

Solution:

Equation is in standard form.

Factor the quadratic expression:

(x + 6) (x – 2) = 0.

Now set each factor equal to zero:

x + 6 = 0

x – 2 = 0

Solve for

x = -6 or x = 2

The solution are x = -6 and x = 2

Practice Questions on Quadratic equations by factoring

Question 1: Solve x²+ 7x + 10 = 0 , by using factoring method

Solution:

First, we check equation is in standard form.

Now we factor in the quadratic expression:

x² + 7x + 10 = 0

(x + 2) (x + 5) = 0

Now set each factor equal to zero:

x + 2 = 0 or x + 5 = 0

Now we solve for x:

x = -2 or x = -5

The solution for the above quadratic equation is x = -2 and x = -5.

Question 2: Solve x² − 5x + 6 = 0 by using the factoring method.

Solution:

First, check that the equation is in standard form.

Now, factor the quadratic expression:

x² − 5x + 6=0

(x − 2) (x − 3) = 0

Now, set each factor equal to zero:

x − 2 = 0 or x − 3 = 0

Now, solve for x:

x = 2 or x = 3

The solution for the above quadratic equation x = 2 and x = 3.

Question 3: Solve 2x² + 7x + 3 = 0 by using the factoring method.

Solution:

First, check that the equation is in standard form.

Now, factor the quadratic expression:

2x² + 7x + 3 = 0

(2x + 1)(x + 3) = 0

Now, set each factor equal to zero:

2x + 1 = 0 or x + 3 = 0

Now, solve for x:

2x = −1 → x = -1/2

x = −3

The solution for the above quadratic equation is x = -1/2 and x = −3.

Question 4: Solve x² + 3x − 4 = 0 by using the factoring method.

Solution:

First, check that the equation is in standard form.

Now, factor the quadratic expression:

x² + 3x − 4 = 0

(x + 4) (x − 1) = 0

Now, set each factor equal to zero:

x + 4 = 0 or x − 1 = 0

Now, solve for x:

x = −4 or x = 1
The solution for the above quadratic equation is x = −4 and x = 1.

Question 5: Solve 3x² − 2x − 8 = 0 by using the factoring method.

Solution:

First, check that the equation is in standard form.

Now, factor the quadratic expression:

3x² − 2x − 8 = 0

(3x + 4) (x − 2) = 0

Now, set each factor equal to zero:

3x + 4 = 0 or x − 2 = 0

Now, solve for x:

3x = −4

x = -4/3

x = 2

The solution for the above quadratic equation x = -4/3 and x = 2.

Question 6: Solve x² − 3x −10 = 0 by using the factoring method.

Solution:

First, check that the equation is in standard form.

Now, factor the quadratic expression:

x² − 3x − 10 = 0

(x − 5) (x + 2) = 0

Now, set each factor equal to zero:

x − 5 = 0 or x + 2 = 0

Now, solve for x:

x = 5 or x = −2

The solution for the above quadratic equation is x = 5 and x = −2.

Conclusion

Factoring is one of important method to solve quadratic equations. Quadratic equations are very useful in various fields, and mastering their solutions is crucial for getting good at academics. When used appropriately, this factoring approach improves problem-solving skills. Through learning and application of the various factoring methods, students can get a good command of quadratic equation and a good hold of algebra.

Read More:

• Roots of Quadratic Equation
• Solving Quadratic Equations
• Nature of Roots

Frequently Asked Questions

What is factoring?

Factoring in algebra is the process of breaking down a polynomial function to simplify it so that it can be understood much better. When these simplified equations are combined it give back polynomial function as a result.

Is factoring an effective way to solve all quadratic equations?

No, factoring is not a solution for every quadratic equation. Certain quadratics necessitate alternative approaches, like completing the square or applying the quadratic formula, as they don’t fold nicely into integers.

Which methods of factoring quadratic equations are commonly used?

Factoring by common factors, grouping, factoring trinomials, factoring perfect square trinomials, and factoring the difference of squares are examples of frequent factoring approaches.

What do factored quadratic equation solutions represent?

The x-values where the quadratic expression equals zero are represented by the solutions of a factored quadratic equation. These are the locations where the x-axis and the quadratic equation’s graph cross.

What do the solutions of a factored quadratic equation represent?

The solutions of a factored quadratic equation represent the x-values where the quadratic expression equals zero. These are the points where the graph of the quadratic equation intersects the x-axis.