Solving Quadratic Equations by Factoring

Solving Quadratics by Factoring

To solve a quadratic equation using factoring :

• Transform the equation using the standard form in which one side is zero.
• Factor the non-zero side.
• Set each factor to zero (Remember: a product of factors is zero if and only if one or more of the factors is zero).
• Solve each resulting equation.

Example 1:

Solve the equation, x2−3x−10=0

Factor the left side: (x−5)(x+2)=0

Set each factor to zero: x−5=0  or  x+2=0

Solve each equation: x=5  or  x=−2

The solution set is {5,−2}.

Example 2:

Solve the equation, 2x2+5x=12

Set the right side to zero: 2x2+5x−12=0

Factor the left side: (2x−3)(x+4)=0

Set each factor to zero: 2x−3=0 or x+4=0

Solve each equation: x=32 or x=−4

The solution set is {32,−4} .

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What Is Factoring Quadratics?

Factoring is often the quickest and most straightforward way of solving a quadratic problem. In mathematics, factoring is the process of identifying expressions that may be multiplied together to produce the expression on one side of a mathematical equation. It is possible to factor a quadratic equation, and in this case the equation is expressed as a product of linear terms. In quadratic equations, factoring quadratics is the process of expressing a polynomial as a product of its linear factors.

When we simplify quadratic expressions, we may locate their roots and use them to solve equations. A quadratic polynomial is a polynomial of the form ax2 + bx + c, where a, b, and c are all positive integers. Factoring quadratics is a technique that allows us to get the zeros of the quadratic equation ax2 + bx + c = 0 by dividing the equation by the number of factors. Let us learn about the intriguing idea of factoring quadratics in this mini-lesson. We will also learn about the formula for factorization of quadratic equations, as well as some solved examples to help us comprehend the concept better. You can also use the Factoring Calculator tool.

'Factoring quadratics' is an expression for the quadratic equation ax2 + bx + c = 0, which is a product of its linear factors, expressed as (x - k)(x - h). The roots of the quadratic equation are denoted by h and k, while the roots of the quadratic equation are denoted by. This approach is sometimes referred to as the method of factorization of quadratic equations or the factorization method. Factorization of quadratic equations may be accomplished using a variety of ways, including dividing the middle term, applying the quadratic formula, and completing the squares, among others.

Main Idea of Using Factoring Method to Solve a Quadratic Equation

Factoring is a popular mathematical procedure that is used to decompose the factors, or numbers, that multiply together to generate another number into their constituent parts. Some numbers are influenced by a variety of circumstances. For example, when you multiply the factors of 6 and 4, 8 and 3, 12 and 2, and 24 and 1, you get the number 24 as a result. Using factoring to solve a variety of numerical issues may be quite beneficial. You get another factor when you multiply a number by one of its factors.

For example, the number 24 divided by the factor 3 provides the number 8. If you have an eight-piece pie that you want to divide among four individuals, factoring may assist you in determining that each person should get two pieces of the pie. The sum of eight pieces divided by four persons equals two pieces for each individual. Alternatively, four persons multiplied by two pieces each person results in eight pieces. If you want to solve quadratic equations easily then Quadratic Equation Solver tool.

Factoring and Solving a Quadratic Equation of Higher Order

When solving a higher degree polynomial, the aim is the same as when solving a quadratic or a basic algebraic expression: factor the polynomial as much as possible, then utilise the factors to discover solutions to the polynomial at the zero point. There are a variety of techniques to solving polynomials with an x3x3 term or greater degree of display style. It's possible that you'll have to try numerous different approaches before you discover one that works for you. The terms ratio and rate relate to two fundamental mathematical notions.

A ratio is a comparison between two numbers or quantities, and it is often expressed as a fraction or decimal. A ratio of three cats to two dogs, for example, may be represented as "3:2" if a person owns three cats and two dogs. "Three to two" is the correct way to interpret this. A rate is a form of ratio in which two distinct units of measurement are compared to one another. For example, if a person runs three miles in 30 minutes, he runs at a pace of one mile every ten minutes throughout that time period.

Examples of How to Solve Quadratic Equations Using the Factoring Method

Factoring quadratics gives us the roots of the quadratic equation. There are different methods that can be used for factoring quadratic equations. Factoring quadratics is done in 4 ways:

• Factoring out the GCD
• Splitting the middle term
• Using Algebraic Identities (Completing the Squares)
• Using Quadratic formula

Factoring quadratics can be done by finding the common numeric factor and the algebraic factors shared by the terms in the quadratic equation and then take them out. Let us solve an example to understand the factoring quadratic equations by taking the GCD out.

Consider this quadratic equation: 3x2 + 6x = 0

The numerical factor is 3 (coefficient of x2) in both terms.

The algebraic common factor is x in both terms.

The common factors are 3 and x. Hence we take them out.

Thus 3x2 + 6x = 0 is factorized as 3x(x + 2) = 0

We split the middle term b of the quadratic equation ax2 + bx + c = 0 when we try to factorize quadratic equations. We determine the factor pairs of the product of a and c such that their sum is equal to b.

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