CBSE 10
Boost Your Math Scores with the Ultimate Quadratic Equations Guide: Unlock the Secrets to Success with Rohan’s Guide
May 18, 2024
Aha
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As a student, it is important for you to have a strong understanding of quadratics (quadratic equations). This topic includes the definition of quadratic equations, the standard form of quadratic equations, the quadratic formula, methods to solve quadratic equations, and various related concepts. By the end of your study, you will be equipped with problem-solving skills that will help you excel in your academic journey.
A quadratic equation is a second-degree polynomial equation in one variable (typically represented as "x") that can be written in the form "ax2 + bx + c = 0". The coefficients "a", "b", and "c" are real numbers, and "a" should not be equal to 0.
The standard form of a quadratic equation is "ax2 + bx + c = 0", where "a", "b", and "c" are coefficients. This form helps in identifying the leading coefficient, linear coefficient, and constant term of the quadratic equation.
The quadratic formula is a powerful tool used to find the solutions (roots) of a quadratic equation. It is given by:
x = (-b ± √(b2 - 4ac)) / (2a)
Here, "a", "b", and "c" are the coefficients of the quadratic equation. The "±" symbol represents both the positive and negative values of the square root.
Let's look at a few examples of quadratic equations:
Solve the quadratic equation x2 - 4x + 4 = 0.
✅ Solution:
To solve the quadratic equation x2 - 4x + 4 = 0, various methods can be used such as factoring, completing the square, or the quadratic formula. Let's use the factoring method in this case.
The given equation, x2 - 4x + 4 = 0, is a perfect square trinomial. It can be factored as:
(x - 2)2 = 0.
Setting the expression equal to zero, you have:
(x - 2)2 = 0.
Now, to find the solutions, you can take the square root of both sides:
√((x - 2)2) = √0.
(x - 2) = 0.
Solving for x:
x = 2.
Therefore, the solution to the quadratic equation x2 - 4x + 4 = 0 is x = 2.
Solve the quadratic equation 2x2 + 3x - 2 = 0.
✅ Solution:
To solve the quadratic equation 2x2 + 3x - 2 = 0, you can use various methods such as factoring, completing the square, or the quadratic formula. Let's use the quadratic formula in this case.
The quadratic formula is given by:
x = (-b ± √(b2 - 4ac)) / (2a).
Comparing the given equation with the standard form ax2 + bx + c = 0, you can identify the values of a, b, and c as follows:
a = 2b = 3c = -2.
Substituting these values into the quadratic formula:
x = (-(3) ± √((3)^2 - 4(2)(-2))) / (2(2)).
Simplifying further:
x = (-3 ± √(9 + 16)) / 4,
x = (-3 ± √25) / 4.
For the "+" case:
x = (-3 + 5) / 4,
x = 2 / 4,
x = 1/2.
For the "-" case:
x = (-3 - 5) / 4,
x = -8 / 4,
x = -2.
Therefore, the solutions to the quadratic equation 2x2 + 3x - 2 = 0 are x = 1/2 and x = -2.
There are several methods to solve quadratic equations. Let's explore some commonly used methods:
If a quadratic equation can be factored into two binomials, equating each factor to zero helps find the solutions. This method is useful when the equation can be easily factored.
Example: Solve the quadratic equation x2 - 7x + 10 = 0.
Solution: Factoring gives (x - 5)(x - 2) = 0. Setting each factor to zero, you’ll find x = 5 and x = 2 as the solutions.
The completing the square method involves transforming a quadratic equation into a perfect square trinomial. This method is useful when factoring becomes challenging.
Example: Solve the quadratic equation x2 + 6x - 4 = 0.
Solution: Completing the square gives (x + 3)2 - 13 = 0. Rearranging, you have (x + 3)2 = 13. Taking the square root, you get x + 3 = ±√13. Thus, the solutions are x = -3 + √13 and x = -3 - √13.
As mentioned earlier, the quadratic formula is a reliable method to find the solutions of a quadratic equation. It can be used for any quadratic equation, regardless of its factorability.
Example: Solve the quadratic equation 3x2 - 2x + 1 = 0.
Solution: By substituting the coefficients into the quadratic formula, you’ll find x = (2 ± √(22 - 4(3)(1))) / (2(3)). Simplifying, you get x = (2 ± √(-8)) / 6. Since the square root of a negative number is involved, the solutions are complex: x = (2 ± 2i√2) / 6.
This method works well when the quadratic equation is a perfect square.
Example: Solve the quadratic equation 4x2 = 36.
Solution: Taking the square root of both sides, you’ll have √(4x2) = √36. Simplifying, you get 2x = ±6. Dividing by 2, you’ll find x = ±3 as the solutions.
These methods provide flexibility in solving different types of quadratic equations based on their characteristics.
Solve the quadratic equation 2x2 - 5x - 3 = 0.
✅ Solution:
To solve this equation, you can use the quadratic formula:
x = (-b ± √(b2 - 4ac)) / (2a).
Comparing the given equation with the standard form ax2 + bx + c = 0, you can identify the values of a, b, and c as follows:
a = 2b = -5c = -3.
Now, substituting these values into the quadratic formula:
x = (-( -5 ) ± √((-5)2 - 4(2)(-3))) / (2(2)).
Simplifying further:
x = (5 ± √(25 + 24)) / 4,
x = (5 ± √49) / 4,
x = (5 ± 7) / 4.
For the "+" case:
x = (5 + 7) / 4,
x = 12 / 4,
x = 3.
For the "-" case:
x = (5 - 7) / 4,
x = -2 / 4,
x = -1/2.
Therefore, the solutions to the quadratic equation 2x^2 - 5x - 3 = 0 are x = 3 and x = -1/2.
Find the roots of the quadratic equation x2 + 7x + 10 = 0.
✅ Solution:
To solve this equation, you can use factoring:
(x + 5)(x + 2) = 0.
Setting each factor to zero:
x + 5 = 0 or x + 2 = 0.
Solving for x in each case:
For x + 5 = 0:x = -5.
For x + 2 = 0:x = -2.
Therefore, the solutions to the quadratic equation x2 + 7x + 10 = 0 are x = -5 and x = -2.
Solve the quadratic equation 3x2 + 4x = 1.
✅ Solution:
To solve the quadratic equation 3x2 + 4x = 1, you can rearrange it into the standard form by moving the constant term to the left side:
3x2 + 4x - 1 = 0.
Now, you can use the quadratic formula to find the solutions:
x = (-b ± √(b2 - 4ac)) / (2a).
Comparing the given equation with the standard form ax2 + bx + c = 0, you can identify the values of a, b, and c as follows:
a = 3b = 4c = -1.
Substituting these values into the quadratic formula:
x = (-(4) ± √((4)2 - 4(3)(-1))) / (2(3)).
Simplifying further:
x = (-4 ± √(16 + 12)) / 6,
x = (-4 ± √28) / 6.
To simplify the expression inside the square root:
√28 = √(4 * 7) = 2√7.
Substituting this back into the equation:
x = (-4 ± 2√7) / 6.
Simplifying the expression further:
Dividing both the numerator and denominator by 2, we obtain x = (-2 ± √7) / 3.
Therefore, the solutions to the quadratic equation 3x^2 + 4x = 1 are x = (-2 + √7) / 3 and x = (-2 - √7) / 3.
Find the solutions of the quadratic equation 5x2 - 9 = 0.
✅ Solution:
To solve this equation, you can isolate x^2 by moving the constant term to the right side:
5x2 = 9.
Next, you divide both sides by 5 to obtain:
x2 = 9/5.
To find the square root of both sides, consider both the positive and negative square roots:
x = ±√(9/5).
Simplifying further:
x = ±(√9/√5),
x = ±(3/√5).
To rationalize the denominator, multiply both the numerator and denominator by √5:
x = ±(3√5/√(5*5)),
x = ±(3√5/5).
Therefore, the solutions to the quadratic equation 5x2 - 9 = 0 are x = 3√5/5 and x = -3√5/5.
Here are answers to some frequently asked questions related to quadratics:
A quadratic equation is a second-degree polynomial equation in one variable, represented as
"ax2 + bx + c = 0".
The methods to solve a quadratic equation include factoring, completing the square, using the quadratic formula, and taking the square root.
Yes, x2 - 1 is a quadratic equation since it can be written in the form "ax^2 + bx + c = 0".
The solutions of the equation x2 + 4 = 0 are x = 2i and x = -2i, where "i" represents the imaginary unit (√(-1)).
Yes, the quadratic equation can be expressed in the form (x - α)(x - β) = 0, where α and β are the roots of the equation.
Understanding quadratics is essential in various fields, including mathematics, physics, and engineering. Mastering the concepts and techniques associated with quadratic equations will empower you to solve complex problems and excel in your academic journey. So, dive into the world of quadratics, practice solving equations, and unlock your problem-solving potential.
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