Introduction & Solving Quadratic Equations

# Introduction & Solving Quadratic Equations | Mathematics (Maths) Class 10 PDF Download

Quadratic Equations occur almost everywhere in our real life. For example, even the problem of designing a playground can be formulated as a quadratic equation. When so many situations give rise to quadratic equations, it sparks a genuine interest in looking for their solutions. Let’s say Q(x) = 0 is a quadratic equation. The solutions to a quadratic equation represent the points where this equation is satisfied that is Q(x) = 0. The solutions are also called roots/zeros of the quadratic equation. Let’s look at some approaches for solving the quadratic equations.

### Quadratic Equation

A quadratic equation is a second-degree polynomial. Its general form is given by,
ax2 + bx + c = 0
a, b and c are real numbers while a ≠ 0. Its shape is a parabola that opens upwards or downwards depending upon the value of “a”.
Its solution is the point where the equation is satisfied. There are several methods of finding out a solution to the quadratic equation given as follows:

• Factoring
• Quadratic Formula

Methods to solve quadratic equations

### Factoring

We try to factor out the equation such that we get the equation in form of the product of two terms. Then on equating these two terms to zero, we get the roots.

The following steps must be used for finding the roots with factorization:

The following steps must be used for finding the roots with factorization:

• All the terms must be on one side of the equation, either LHS or RHS leaving zero on the other side.
• Factorize the equation
• Set the factors equal to zero to find the roots one by one.

Let’s look at this method in more detail using the examples below:
Question 1: 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

Question 2: Factorize the following equation and find its roots: x+ x – 12 = 0
Solution:
x2 + x – 12 = 0
⇒ x2 + 4x – 3x – 12 = 0
⇒ x(x + 4) -3(x + 4) = 0
⇒ (x – 3) (x + 4) = 0
Equating both of these terms with zero.
x – 3 = 0 and x – 4 = 0
x = 3 and 4

### Quadratic Formula

All the quadratic equations can be solved using the quadratic formula.
For an equation of the form,
ax2 + bx + c = 0,
Where a, b and c are real numbers and a ≠ 0.
The roots of this equation are given by,
x =
Given that b2 – 4ac is greater than or equal to zero.

Question 1: Find out the roots of the equation using Quadratic Formula,
Solution:
4x2 + 10x + 3 = 0
Using Quadratic Formula to solve this,
a = 4, b = 10 and c = 3
Before plugging in the values, we need to check for the discriminator
b2 – 4ac
⇒ 102 – 4(4)(3)
⇒ 100 – 48
⇒ 52
This is greater than zero, So now we can apply the quadratic formula.

Plugging the values into quadratic equation,

Question 2: Find out the roots of the equation using Quadratic formula,
5x2 + 9x + 4 = 0
Solution:
5x2 + 9x + 4 = 0
Using Quadratic Formula,
a = 5, b = 9 and c = 4.
Before plugging in the values, we need to check for the discriminator
b2 – 4ac
⇒ 92 – 4(5)(4)
⇒ 81 – 80
⇒ 1
This is greater than zero, So the quadratic formula can be applied. Plugging in the values in the formula,

The document Introduction & Solving Quadratic Equations | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## Mathematics (Maths) Class 10

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## FAQs on Introduction & Solving Quadratic Equations - Mathematics (Maths) Class 10

 1. What is a quadratic equation?
Ans. A quadratic equation is a polynomial equation of degree 2, which means the highest power of the variable is 2. It can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.
 2. How do you solve a quadratic equation by factoring?
Ans. To solve a quadratic equation by factoring, follow these steps: 1. Write the equation in the form ax^2 + bx + c = 0. 2. Factor the quadratic expression on the left side of the equation. 3. Set each factor equal to zero and solve the resulting linear equations. 4. Check the solutions by substituting them back into the original equation.
 3. Can all quadratic equations be solved by factoring?
Ans. No, not all quadratic equations can be solved by factoring. Some quadratic equations may have complex roots or cannot be easily factored. In such cases, other methods like completing the square or using the quadratic formula are used to find the solutions.
 4. What is the quadratic formula?
Ans. The quadratic formula is a formula used to solve quadratic equations. It is given by: x = (-b ± √(b^2 - 4ac))/(2a) where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. The ± sign indicates that there are two possible solutions.
 5. How do you solve a quadratic equation using the quadratic formula?
Ans. To solve a quadratic equation using the quadratic formula, follow these steps: 1. Write the equation in the form ax^2 + bx + c = 0. 2. Identify the values of a, b, and c. 3. Substitute these values into the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a). 4. Simplify the equation and calculate the values of x using the plus and minus signs ±. These values represent the solutions to the quadratic equation.

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