Short Trick to Solve Quadratic Equation - GeeksforGeeks (2024)

Quadratic equations are among the many equations found in mathematics that help us solve various real-world problems. Solving a quadratic equation can sometimes be challenging, especially when we are unable to find the roots through factorization. In such cases, we can utilize the method discussed in this article to determine the roots.

Whether the roots are rational numbers, integers, or complex numbers, this technique allows for their easy identification. This method was discovered by mathematician Poh Shen Loh and is straightforward to apply to any quadratic equation.

What is Quadratic Equation?

A quadratic equation is a second-order polynomial equation in a single variable x. It has the general form:

ax² + bx + c = 0

where a, b, and c are constants with a ≠ 0. The quadratic equation is characterized by its highest exponent, which is 2.

Common Methods to Solve Quadratic Equations

There are many methods to solve quadratic equations such as:

• Factoring Method
• Completing the Square
• Using the Quadratic Formula

Let’s discuss these in detail as follows:

Factoring Method

It’s one of the simplest methods to find the roots of a quadratic equation, but it is only applicable if the roots are integers or simple rational numbers. We can find the roots of a quadratic equation using this method as follows:

Step 1: Write the Quadratic Equation in Standard Form i.e., ax² + bx + c = 0.

Step 2: Identify the factor pairs from factors of ac which add up to b.

Step 3: Rewrite the middle term as sum of factors.

Step 4: Regroup the terms, and factor out common terms.

Step 5: Set Each Factor Equal to Zero and solve for x.

Let’s consider an example for better understanding.

Example : Factorize the following equation and find its roots: 2x²– x – 1 = 0

Solution:

2x²– x – 1 = 0

⇒ 2x²-2x + x – 1 = 0

⇒ 2x(x – 1) + 1(x – 1) = 0

⇒ (2x + 1) (x – 1) = 0

For this equation two be zero, either one of these or both of these terms should be zero.

So, we can find out roots by equating these terms with zero.

2x + 1 = 0

x = -1/2

x – 1 = 0

⇒ x = 1

So, we get two roots in the equation.

x = 1 and-1/2.

Completing the Square

In this method, we convert the given equation into a complete square of a binomial and then simplify to find the answer using the the following steps:

Step 1: Start with the standard form of the quadratic equation.

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Step 2: Move the constant term to the other side of the equation.

Step 3: Divide every term by the coefficient of x² to make it 1.

Step 4: Find the number that completes the square and add its square to both sides of equation.

Step 5: Rewrite the left side as a squared binomial.

Step 6: Take the square root of both sides.

Step 7: Solve for x by isolating it.

Read More about Solving Quadratic Equations using Completing the Square Method.

Using the Quadratic Formula

The quadratic formula states that the solutions to the quadratic equation ax² + bx + c = 0 are given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Where a, b, and c are coefficients of the quadratic equation.

Trick to Solve Quadratic Equations

This method for solving quadratic equations discovered by mathematician Poh Shen Loh. Let’s discuss this method.

To solve equation:

ax² + bx + c = 0

Divide the equation by a,

x² + (b/a)x + c/a = 0

• Sum of Roots = -b/a
• Product of Roots = c/a

Let’s assume the roots of the equation are (-b/2a + x) and (-b/2a – x).

(-b/2a + x) (-b/2a – x) = c/a

Solve for x, and we will get our roots.

Let’s consider an example for better understanding.

Example: Solve for x: 2x² + 8x + 6 = 0

Solution:

Given Equation: 2x² + 8x + 6 = 0

Divide the equation by 2,

x² + 4x + 3 = 0

Now, Sum of Roots = -4

and Product of Roots = 3

Let the roots be -2 + x and -2 – x.

(-2 + x)(-2 – x) = 3

⇒ 4 – x² = 3

⇒ 4 – 3 = x²

⇒ x² = 1

⇒ x = ±1

Thus, roots are -1 and -3.

Read More,

• Standard Form of Quadratic Equation
• Nature of Roots of Quadratic Equation
• Relationship between Zeroes and Coefficients of a Polynomial

Solved Examples

Example 1: Solve for x: 3x² + 10x + 7 = 0

Solution:

Given Equation: 3x² + 10x + 7 = 0

Divide the equation by 3,

x² + (10/3)x + 7/3 = 0

Now, Sum of Roots = -10/3

and Product of Roots = 7/3

Let the roots be -10/6 + x and -10/6 – x.

(-10/6 + x)(-10/6- x) = 7/3

⇒ 100/36 – x² = 7/3

⇒ 100/36 – 7/3 = x²

⇒ (100 – 84)/36 = x²

⇒ x² = 16/36 = 4/9

⇒ x = ± 2/3

Thus, roots are -1 and -7/3.

Example 2: Solve for x: 2x² + 5x – 3 = 0

Solution:

Given Equation: 2x² + 5x – 3 = 0

Divide the equation by 3,

x² + (5/2)x – 3/2 = 0

Now, Sum of Roots = -5/2

and Product of Roots = -3/2

Let the roots be -5/4 + x and -5/4 – x.

(-5/4 + x)(-5/4- x) = -3/2

⇒ 25/16 – x² = -3/2

⇒ 25/16 + 3/2 = x²

⇒ (25 + 24)/16 = x²

⇒ x² = 49/16

⇒ x = ± 7/4

Thus, roots are 1/2 and -3.

Practice Problems

Problem: Find the roots of following quadratic equations.

FAQs on Quadratic Equations Tricks

What is the best method to solve a quadratic equation?

The quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

is the most reliable method to solve a quadratic equation.

Can all quadratic equations be factored?

No, only quadratics with rational roots can be factored.

What if the quadratic equation has no real roots?

The discriminant D = b² − 4ac < 0 indicates that complex roots.

How can I check my solutions?

Substitute the solutions back into the original equation to verify their correctness.

Are there any shortcuts to solving quadratic equations?

Yes, factoring or completing the square can be quicker for simpler quadratic equations.