Solving the Quadratic equations by factoring

Factoring the quadratic expression is a method in mathematics for solving the quadratic equations. In this method, splitting the middle term in the quadratic expression is used to factorize (or factorise) the quadratic expression in the quadratic equation.

Now, let’s learn how to solve the quadratic equations by factoring method in mathematics.

Steps

Make sure there is no expression on the right hand side of the equation by moving all terms to the left hand side of the equation and then write the quadratic expression in descending order. Now, use the following steps to factor the quadratic expression in the equation as a product of two linear expressions in one variable.

1. Multiply the constant by the term in which the exponent of the variable is two, and calculate their product.
2. Split the middle term of the quadratic expression as either sum or difference of two terms such that their product is exactly equal to the product of the constant and the first term of the quadratic expression in descending order.
3. Take the common factors out from the terms for converting the quadratic expression into the factor form. Finally, solve the equation by making each linear expression in one variable equals to zero.

Example

Solve $2x^2+7x+6 \,=\, 0$

There is no term in the right hand side of the equation and the quadratic expression in the left hand side of the equation is in descending order. So, we can start the procedure for factoring the quadratic expression.

Evaluate the Product of first term and the constant

$2x^2$ is the first term and $6$ is the constant in the given quadratic equation. Now, find the product of them by the multiplication.

$\implies$ $2x^2 \times 6 \,=\, 12x^2$

Split the middle term for factoring the expression

$7x$ is the middle term in the given quadratic equation $2x^2+7x+6 \,=\, 0$. Try to expand it as either sum or difference of two terms but their product should be equal to the product of the first term $2x^2$ and the constant $6$.

The sign of the constant term $6$ is positive and the sign of the middle term $7x$ is also positive. Hence, the middle term can be expanded possibly in sum form. Mathematically, the term $7x$ can be split in the following ways.

$(1).\,\,\,$ $x+6x \,=\, 7x$

$(2).\,\,\,$ $2x+5x \,=\, 7x$

$(3).\,\,\,$ $3x+4x \,=\, 7x$

Multiply the terms in the sum form to check whether their product is equal to the product of the first term $2x^2$ and the constant $6$.

$(1).\,\,\,$ $x \times 6x \,=\, 6x^2$

$(2).\,\,\,$ $2x \times 5x \,=\, 10x^2$

$(3).\,\,\,$ $3x \times 4x \,=\, 12x^2$

The sum of the terms $x$ and $6x$ is equal to $7x$ but their product is not equal to the $12x^2$. Similarly, the sum of the terms $2x$ and $5x$ is equal to $7x$ but their product is also not equal to the $12x^2$.

However, the sum of the terms $3x$ and $4x$ is equal to the $7x$ and their product is also equal to the $12x^2$, which is the product of the first term $2x^2$ and the constant $6$. Hence, the middle term $7x$ should be expanded as the sum of the terms $3x$ and $4x$ in the quadratic equation $2x^2+7x+6 \,=\, 0$.

$\implies$ $2x^2+3x+4x+6$ $\,=\,$ $0$

Solve the quadratic equation by the factorization

The factorisation method can be used to take the common factor out from the terms in the quadratic expression.

$\implies$ $2x^2+4x+3x+6$ $\,=\,$ $0$

Factorize each term for separating the common factor from the terms in the quadratic expression.

$\implies$ $2x \times x$ $+$ $2 \times 2x$ $+$ $3 \times x$ $+$ $3 \times 2$ $\,=\,$ $0$

$\implies$ $2x \times (x+2)$ $+$ $3 \times (x+2)$ $\,=\,$ $0$

$\implies$ $(x+2) \times (2x+3)$ $\,=\,$ $0$

$\implies$ $(x+2)(2x+3)$ $\,=\,$ $0$

Each factor in the quadratic equation is a binomial but it is also a linear expression in one variable. Now, let’s solve this equation by the zero product property.

$\implies$ $x+2 \,=\, 0$ or $2x+3 \,=\, 0$

$\implies$ $x \,=\, -2$ or $2x \,=\, -3$

$\,\,\,\therefore\,\,\,\,\,\,$ $x \,=\, -2$ or $x \,=\, \dfrac{-3}{2}$

The quadratic expression $2x^2+7x+6$ is equal to zero when $x \,=\, -2$ or $x \,=\, -\dfrac{3}{2}$. Therefore, the solution set for the given quadratic equation is $\bigg\{-2, -\dfrac{3}{2}\bigg\}$.

Thus, a quadratic equation can be solved mathematically by factoring method in mathematics.

Problems

List of the questions on factoring the quadratic equations for practice and solutions with step by step procedure to learn how to factor the quadratic expressions in solving the quadratic equations in mathematics.

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