10 Examples of Quadratic Equation with Solutions

A quadratic equation is a second-order polynomial equation in a single variable x, with the general form:

ax^2 + bx + c = 0

where x is the variable, and a, b, and c are constants. The values of a, b, and c can be any real numbers (positive, negative, or zero). If a is not equal to zero, then the equation is called a quadratic equation. If a is equal to zero, then the equation is not a quadratic equation, but a linear equation.

The most common method for solving a quadratic equation is to use the quadratic formula:

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

This formula gives the values of x that will make the quadratic equation equal to zero. The plus-minus symbol (±) indicates that there are two possible solutions for x, one using the plus sign and one using the minus sign.

It's important to note that the quadratic formula only gives real solutions for x if the discriminant (b^2 - 4ac) is greater than or equal to zero. If the discriminant is less than zero, then the equation has no real solutions.

10 Examples of Quadratic Equation with Solution:

Here are 10 examples of quadratic equations and their solutions:

Example no 1:

x^2 + 5x + 6 = 0

To solve this equation, we can use the quadratic formula:

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

Plugging in the values for a, b, and c gives us:

x = (-5 +/- sqrt(5^2 - 416)) / (2*1)

x = (-5 +/- sqrt(25 - 24)) / 2

x = (-5 +/- sqrt(1)) / 2

x = (-5 +/- 1) / 2

x = (-4)/2, (-6)/2

x = -2, -3

So the solutions to this equation are x = -2 and x = -3.

Example no 2:

x^2 - 7x + 12 = 0

Using the quadratic formula, we get:

x = (-7 +/- sqrt(7^2 - 4112)) / (2*1)

x = (-7 +/- sqrt(49 - 48)) / 2

x = (-7 +/- sqrt(1)) / 2

x = (-7 +/- 1) / 2

x = (-6)/2, (-8)/2

x = -3, -4

So the solutions to this equation are x = -3 and x = -4.

Example no 3:

x^2 + 3x - 4 = 0

Using the quadratic formula, we get:

x = (3 +/- sqrt(3^2 - 41(-4))) / (2*1)

x = (3 +/- sqrt(9 + 16)) / 2

x = (3 +/- sqrt(25)) / 2

x = (3 +/- 5) / 2

x = 8/2, -2/2

x = 4, -1

So the solutions to this equation are x = 4 and x = -1.

Example no 4:

x^2 - 5x + 6 = 0

Using the quadratic formula, we get:

x = (-5 +/- sqrt(5^2 - 416)) / (2*1)

x = (-5 +/- sqrt(25 - 24)) / 2

x = (-5 +/- sqrt(1)) / 2

x = (-5 +/- 1) / 2

x = (-4)/2, (-6)/2

x = -2, -3

So the solutions to this equation are x = -2 and x = -3.

Example no 5:

x^2 + x - 6 = 0

Using the quadratic formula, we get:

x = (1 +/- sqrt(1^2 - 41(-6))) / (2*1)

x = (1 +/- sqrt(1 + 24)) / 2

x = (1 +/- sqrt(25)) / 2

x = (1 +/- 5) / 2

x = 6/2, -4/2

x = 3, -2

So the solutions to this equation are x = 3 and x = -2.

Example no 6:

x^2 - 6x + 9 = 0

Using the quadratic formula, we get:

x = (-6 +/- sqrt(6^2 - 419)) / (2*1)

x = (-6 +/- sqrt(36 - 36)) / 2

x = (-6 +/- sqrt(0)) / 2

x = (-6 +/- 0) / 2

x = -6/2, -6/2

x = -3, -3

So the solutions to this equation are x = -3 and x = -3.

Example no 7:

x^2 + 2x + 7 = 0

Using the quadratic formula, we get:

x = (2 +/- sqrt(2^2 - 411)) / (2*1)

x = (2 +/- sqrt(4 - 4)) / 2

x = (2 +/- sqrt(0)) / 2

x = (2 +/- 0) / 2

x = 2/2, 2/2

x = 1, 1

So the solutions to this equation are x = 1 and x = 1.

Example no 8:

x^2 - 4x + 4 = 0

Using the quadratic formula, we get:

x = (-4 +/- sqrt(4^2 - 414)) / (2*1)

x = (-4 +/- sqrt(16 - 16)) / 2

x = (-4 +/- sqrt(0)) / 2

x = (-4 +/- 0) / 2

x = -4/2, -4/2

x = -2, -2

So the solutions to this equation are x = -2 and x = -2.

Example no 9:

x^2 - x - 6 = 0

Using the quadratic formula, we get:

x = (-1 +/- sqrt(1^2 - 41(-6))) / (2*1)

x = (-1 +/- sqrt(1 + 24)) / 2

x = (-1 +/- sqrt(25)) / 2

x = (-1 +/- 5) / 2

x = 4/2, -6/2

x = 2, -3

So the solutions to this equation are x = 2 and x = -3.

Example no 10:

x^2 + x + 1 = 0

Using the quadratic formula, we get:

x = (1 +/- sqrt(1^2 - 411)) / (2*1)

x = (1 +/- sqrt(1 - 4)) / 2

x = (1 +/- sqrt(-3)) / 2

There are no real solutions to this equation because the square root of a negative number is not a real number.

Practice Questions:

Here are 10 quadratic equations for you to practice solving:

x^2 - 3x - 10 = 0

x^2 + 2x - 8 = 0

x^2 - 5x + 6 = 0

x^2 + 7x + 10 = 0

x^2 - x - 1 = 0

x^2 - 4x + 3 = 0

x^2 + x - 2 = 0

x^2 - x - 3 = 0

x^2 + 2x + 1 = 0

x^2 - 6x + 9 = 0

I hope these equations are helpful for your practice! Let me know if you have any questions.