Choosing a Method to Solve Quadratics: Videos & Practice Problems

To solve quadratic equations effectively, it's essential to choose the best method based on the equation's characteristics. Start with factoring if there are obvious factors or if the constant term is zero. If not, check the square root property for the form x + a² = b or if b equals zero. Completing the square is suitable when the leading coefficient is 1 and b is even. If none of these apply, the quadratic formula is the final option. Understanding these methods enhances problem-solving efficiency in quadratic equations.

Choosing a Method to Solve Quadratics

When solving quadratic equations, selecting the most effective method can significantly simplify the process. While multiple methods may be applicable, understanding the characteristics of the quadratic can guide you to the best choice. Here’s a structured approach to determine which method to use based on the equation's form and coefficients.

Start by examining the equation. If the quadratic is in the form ax² + bx + c = 0, consider the following methods:

Factoring: This method is ideal when the equation has obvious factors or when the constant term c equals zero. For example, in the equation x² + 3x = 0, since c = 0, factoring is the best approach.

Square Root Property: This method is applicable when the equation can be expressed as (x + p)² = q or when the coefficient b equals zero. For instance, in the equation (x + 2)² = 9, the square root property is the most straightforward method to apply.

Completing the Square: This method works best when the leading coefficient a is 1 and b is an even number. For example, in the equation x² + 6x + 1 = 0, since the leading coefficient is 1 and b = 6 (even), completing the square is the optimal choice.

Quadratic Formula: If none of the previous methods are suitable, particularly when the leading coefficient is not 1 or when factoring is not feasible, the quadratic formula x = \frac{-b \pm \sqrt{b² - 4ac}}{2a} can be used. For example, in the equation 2x² + 7x + 3 = 0, since it does not factor easily and the leading coefficient is not 1, the quadratic formula is the best method to solve it.

By systematically evaluating the characteristics of the quadratic equation, you can effectively choose the most efficient method for solving it. This approach not only streamlines the solving process but also enhances your understanding of the relationships between the coefficients and the methods available.

Choose and apply the best method to solve the given quadratic equation.
$x^2-6x=5$

• A. $x=6+\sqrt{14},x=6-\sqrt{14}$
• B. $x=10,x=-4$
• C. $x=3+\sqrt{14},x=3-\sqrt{14}$
• D. $x=6,x=0$

Answer: (C)

Choose and apply the best method to solve the given quadratic equation. 
$4x^2+16x+12=0$

• A. $x=-3,x=-1$
• B. $x=3,x=1$
• C. $x=3,x=4$
• D. $x=-12,x=1$

Answer: (A)

Here’s what students ask on this topic:

What is the best method to solve a quadratic equation when the constant term is zero?

When the constant term (C) in a quadratic equation is zero, factoring is often the best method to solve it. This is because the equation can be easily factored into two binomials, allowing you to find the roots by setting each factor equal to zero. For example, if you have an equation like $x^2+\; 3x\; =\; 0$, you can factor it as $x(x+\; 3)\; =\; 0$. Setting each factor to zero gives the solutions $x=\; 0$ and $x=\; -3$. This method is straightforward and efficient when C is zero.

How do you choose the best method to solve a quadratic equation?

Choosing the best method to solve a quadratic equation involves analyzing the equation's structure and coefficients. Start with factoring if the equation has obvious factors or if the constant term (C) is zero. If the equation is in the form $(x+\mathrm{number})2^{}=\mathrm{constant}$, use the square root property. Completing the square is effective when the leading coefficient (A) is one and the coefficient of X (B) is even. If none of these methods fit, use the quadratic formula, especially when the leading coefficient is not one. Practice helps in quickly identifying the best method.

When is completing the square the best method to solve a quadratic equation?

Completing the square is the best method to solve a quadratic equation when the leading coefficient (A) is one and the coefficient of X (B) is even. This method involves rewriting the equation in the form $(x+\mathrm{number})2^{}=\mathrm{constant}$, which makes it easier to solve by taking the square root of both sides. For example, if you have $x^2+\; 6x\; +\; 1\; =\; 0$, you can complete the square to find the roots. This method is particularly useful for equations that do not factor easily and when the quadratic formula seems cumbersome.

Why is the quadratic formula used when other methods are not suitable?

The quadratic formula is used when other methods are not suitable because it provides a universal solution for any quadratic equation, regardless of its structure. It is particularly useful when the leading coefficient (A) is not one, or when the equation does not fit the criteria for simpler methods like factoring, square root property, or completing the square. The formula is $x=\frac{-b\pm \sqrt{b^2-\; 4ac}}{2a}$, and it allows you to find the roots of any quadratic equation by substituting the coefficients into the formula. This method is reliable and works for all types of quadratic equations.

What is the square root property in solving quadratic equations?

The square root property is a method used to solve quadratic equations that are in the form $(x+\mathrm{number})2^{}=\mathrm{constant}$. This property allows you to take the square root of both sides of the equation to solve for X. It is particularly useful when the coefficient of X (B) is zero, simplifying the equation to a form where the square root can be directly applied. For example, if you have $(x+\; 2)2^{}=\; 9$, you can solve it by taking the square root of both sides, resulting in $x+\; 2\; =\; \pm 3$, and then solving for X. This method is efficient for equations that fit this specific form.