Quadratic equations are algebraic expressions of order ‘2’. They are categorised amongst equations of higher orders. The topic is very common in all the competitive exams and requires a lot of practice from young learners. But, what are these equations and where can we use them? What are the tips for solving quadratic equations? Refer below for an in-depth explanation.
Table of Contents:
- What are Quadratic Equations?
- Quadratic Equation Formula/ Formulas for Solving Quadratic Equations
- Roots of Quadratic Equation
- Nature of Roots of quadratic equation
- Relationship Between Coefficients and Roots of Quadratic Equation
- Methods to Solve Quadratic Equations
- Solving a Quadratic Equation – Tips and Tricks
- Examples on Quadratic Equations
- FAQs on Quadratic Equation
1. What are Quadratic Equations?
The word quadratic in the term Quadratic equations is derived from quadratus, a Latin word for ‘square.’ Hence, we define quadratic equations as equations where the variable is of the second degree. Therefore, they are also called “Equations of degree 2”.
When we take a peek into the history, practical applications of quadratic equations are dated way back, as early as 2050 BC. It’s a known fact that mathematicians from Babylonia, Egypt, Greece, China, and India used geometric methods to solve quadratic equations. Over time, various historians kept innovating new ways of finding new formulas for solving quadratic equations.
In 12th century, Spain, a Jewish mathematician Abraham bar Hiyya Ha-Nasi authored the first European book that included a complete solution to the general quadratic equation and finally in 1637, René Descartes, published La Géométrie that contained all the quadratic equation formula that we know today.
2. Quadratic Equation Formula/ Formulas for Solving Quadratic Equations
Formulas for solving quadratic equations provide students with requisite knowledge to deal with complex numericals easily. Below-mentioned is the general quadratic equation formula. We define it as follows:
If ax2 + bx + c = 0 is a quadratic equation, then the value of x is given by the following formula:
Quadratic Equation Formula:
Just plug in the values of a, b and c, and do the calculations. The quantity in the square root is called the discriminant or D. Here’s an example that will give you an understanding on what it takes while solving quadratic equations.
Example: Solve: x2 + 2x + 1 = 0
Solution: Given that a=1, b=2, c=1, and
Discriminant = b2 − 4ac = 22 − 4×1×1 = 0
Using the quadratic equation formula, x = (−2 ± √0)/2 = −2/2
Therefore, x = − 1
3. Roots of Quadratic Equation
Students often wonder if a quadratic equation can have more than one solution? Are there any equations that don’t have any real solution? Yes, it’s possible with the root of quadratic equation concept.
The value of a variable for which the equation gets satisfied is called the solution or the root of quadratic equation. The number of roots of a polynomial equation is equal to its degree. Hence, a quadratic equation has 2 roots. Let α and β be the roots of quadratic equation in the general form: ax2 + bx + c = 0. The formulas for solving quadratic equations can be write as:
(-b-√b2-4ac)/2a and (-b+√b2-4ac)/2a
Here a, b, and c are real and rational. Hence, the nature of the roots α and β of equation ax2 + bx + c = 0 depends on the quantity or expression (b2 – 4ac) under the square root sign. We say this because the root of a negative number can’t be any real number. Say x2 = -1 is a quadratic equation. There is no real number whose square is negative. Therefore for this equation, there are no real number solutions.
Formulas for solving quadratic equations
Hence, the expression (b2 – 4ac) is called the discriminant of the quadratic equation ax2 + bx + c = 0. Its value determines the nature of the roots of quadratic equation.
4. Nature of Roots of Quadratic Equation
Nature of Roots of Quadratic Equation Cases:
Let us recall the general formulas for solving quadratic equations, α = (-b-√b2-4ac)/2a and β = (-b+√b2-4ac)/2a
Case I: b2 – 4ac > 0
When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive, then the roots α and β of the quadratic equation formula ax2+bx+ c = 0 are real and unequal.
Case II: b2– 4ac = 0
When a, b, and c are real numbers, a ≠ 0 and the discriminant is zero, then the roots α and β of the quadratic equation ax2+ bx + c = 0 are real and equal.
Case III: b2– 4ac < 0
When a, b, and c are real numbers, a ≠ 0 and the discriminant is negative, then the roots α and β of the quadratic equation formula ax2 + bx + c = 0 are unequal and not real. In this case, we say that the roots are imaginary.
Case IV: b2 – 4ac > 0 and perfect square
When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real, rational and unequal.
Case V: b2 – 4ac > 0 and not perfect square
When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive but not a perfect square then the roots of the quadratic equation formula ax2 + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.
Case VI: b2 – 4ac > 0 is perfect square and a or b is irrational
When a, b, and c are real numbers, a ≠ 0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax2 + bx + c = 0 are irrational.
Nature of Roots of Quadratic Equation Table Format Summary:
|b2 – 4ac > 0||Real and unequal|
|b2 – 4ac = 0||Real and equal|
|b2 – 4ac < 0||Unequal and Imaginary|
|b2 – 4ac > 0 (is a perfect square)||Real, rational and unequal|
|b2 – 4ac > 0 (is a perfect square)||Real, irrational and unequal|
|b2 – 4ac > 0 (is a perfect square and a or b is irrational)||Irrational|
Nature of Roots of Quadratic Equation Examples:
Example: Discuss the nature of the roots of quadratic equation 2×2 – 8x + 3 = 0.
Solution: Here the coefficients are all rational. The discriminant D of the given equation is
D = b2 – 4ac = (-8)2 – 4 x 2 x 3
= 64 – 24
= 40 > 0
Clearly, the discriminant of the given quadratic equation is positive but not a perfect square. Therefore, the nature of roots of quadratic equation are real, irrational and unequal.
5. Relationship Between Coefficients and Roots of Quadratic Equation
Let’s pick equation ax2 + bx + c = 0. It is one of the formulas for solving quadratic equations, where (a ≠ 0)
Assume α and β to be the roots of the equation ax2 + bx + c = 0
Sum of the roots = α + β = -b/a = (- coefficient of x/ coefficient of x2).
Product of the roots = α x β = c/a = (constant term/ coefficient of x2).
Difference of the roots =
(α – β)2= (α + β)2 – 4αβ = b2/a2 – 4c/a = (b2-4ac)/a2 = (√b2-4ac)/a = √D/a
The quadratic equation can always be expressed in terms of sum and product of roots = x2-(Sum of roots) x + product of roots
= x2 – (α + β)x + αβ = (x-α) (x-β)
6. Methods to Solve Quadratic Equations
Unique methods help in deriving solutions to complex sums and equations. While solving quadratic equations too, there are 3 algebraic methods and 1 graphical method that are beneficial. They are:
- Completing the square method
- Quadratic Equation Formula
In addition to the three methods discussed here, we also have a graphical method. As you may have guessed, it involves plotting the given equation for various values of x. The intersection of the curves thus obtained with the real axis will give us the solutions.
The first and simplest method of solving quadratic equations is the factorization method. Certain algebraic expressions can be factored. These factors, if done correctly, will give two linear equations in x. Hence, from these equations, we get the value of x.
- Completing the Square Method/ Method of Completing the Square
Each quadratic equation has a square term. If we could get two square terms on two sides of the quality sign, we will again get a linear equation. Let us see an example first and understand how we go about solving quadratic equations.
Example: Let us consider the equation, 2×2=12x+54. Here’s how to solve a quadratic equation, step by step by completing the square.
Solution: Let us write the equation 2×2=12x+54. In the standard form, we can write it as: 2×2 – 12x – 54 = 0. Next let us get all the terms with x2 or x in them to one side of the equation: 2×2 – 12 = 54
In the next step, we have to make sure that the coefficient of x2 is 1. So dividing through by the coefficient of x2, we have: 2×2/2 – 12x/2 = 54/2 or x2 – 6x = 27. Next, we make the left hand side a complete square by adding (6/2)2 = 9 i.e. (b/2)2 where ‘b’ is the new coefficient of ‘x’, to both sides as: x2 – 6x + 9 = 27 + 9 or x2 – 2×3×x + 32 = 36. Now we can write it as a binomial square:
- (x-3)2 = 36; Take square root of both sides
- x – 3 = ±6; Which gives us these equations:
- x = (3+6) or x = (3-6) or x = 9 or x = -3
This is known as the method of completing the squares.
If you want to learn more about solving quadratic equations, click here.
7. Solving Quadratic Equations – Tips and Tricks
Many students dedicate hours in solving quadratic equations but still are not able to secure high marks in the exams. The primary reason for this is improper preparation strategy and too much focus on memorising quadratic equation formulas.
Students need a well-structured study plan that helps them achieve desired learning outcomes. Here are some tips and tricks that will help in solving quadratic equations of higher order within no time.
- Understand the sign roots and memorise them:
|Sign of coefficient ‘x’||Sign of coefficient ‘y’||Sign of roots|
- Practice unique questions instead of solving same type of questions multiple times
- Note down your question-solving time and aim to reduce it as much as possible to improve efficiency
- Learn to find out factors as quickly as possible to solve numerical problems in exams swiftly
- Take regular breaks to break free from the monotony and to refresh your mind
8. Examples on Quadratic Equations
Learning formulas for solving quadratic equations and understanding the nature of roots of quadratic equations will set the right base for unraveling even complex equations. Here are some examples on quadratic equations that will bolster your conceptual understanding.
Question 1: Sum of a number and its reciprocal is 5 1/5 . Then the required equation is:
y2 + 1/y = 265
5y2 – 26/y + 5 = 0
y2 + 1/y + 26/5 = 0
5y2 + 26y + 5 = 0
Solution: B) Let ‘y’ be the number. Then we can write y + 1/y = 5 1/5
Therefore, (y2+1)/y = 26/5
y2 + 1 = 26y/5
Hence, 5y2 + 5 – 26y = 0 is the required equation.
Question 2: Solve the equation: x2 + 3x – 4 = 0
Solution: This method is also known as splitting the middle term method. Here, a = 1, b = 3, c = -4. Let us multiply a and c = 1 * (-4) = -4. Next, the middle term is split into two terms. We do it such that the product of the new coefficients equals the product of a and c.
We have to get 3 here. Consider (+4) and (-1) as the factors, whose multiplication is -4 and sum is 3. Hence, we write x2 + 3x – 4 = 0 as x2 + 4x – x – 4 = 0. Thus, we can factorise the terms as: (x+4)(x-1) = 0. For any two quantities a and b, if a×b = 0, we must have either a = 0, b = 0 or a = b = 0.
Thus we have either (x+4) = 0 or (x-1) = 0 or both are = 0. This gives x+4 = 0 or x-1 = 0. Solving these equations for x gives: x=-4 or x=1.
Question 3: Find the value of x: 27x2 − 12 = 0
- A) 2/3 B) ± 2/3 C) Ambiguous D) None of these
Answer : B) Here, a = 27, b = 0 and c = -12. Hence, from the quadratic equation formula, we have:
x = − 0 ± √02 – 4(27)(-12)/2 (27)
Thus x = ± √(4/9) = ± 2/3
Question 4: Without solving, examine the nature of roots of quadratic equation 4×2 – 4x + 1 = 0?
Solution: The discriminant D of the given equation is
D = b2 – 4ac
= (-4)2 – (4 x 4 x 1)
Clearly, the discriminant of the given quadratic equation is zero. Therefore, the roots are real and equal. Hence, here we have understood the nature of roots of quadratic equations very clearly.
9. FAQs on Quadratic Equation
Question 1: Give a suitable example for solving quadratic equations?
Answer: A quadratic equation is the equation of the 2nd degree. This means that it comprises at least one (1) term that is squared. One of the standard formulas for solving quadratic equations is ‘ax² + bx + c = 0’ here a, b, and c are constants or numerical coefficients. ‘X’ here is an unknown variable.
Question 2: What is the use of the quadratic equations?
Answer: The quadratic equations are actually used by us in our daily lives. We use the equations while calculating the area, determining a product’s profit, and formulating the speed of some object.
Question 3: Why are quadratic equations important?
Answer: Actually, the quadratic equation has many purposes in the scientific and mathematical world. Solving quadratic equations is mostly helpful for finding out the curve on a Cartesian grid. It is mainly used for finding the curve that objects take during the time when they fly through the air.
Question 4: How many types of quadratic equations are there?
Answer: There are 3 types of Quadratic Equations:
y = ax2 + bx + c y = (ax + c)(bx + d)
y = a(x + b)2 + c
Question 5: What makes a problem quadratic?
Answer: It describes a problem that deals with a variable multiplied by itself, which we know as squaring. Moreover, this language originates from the area of a square as its side length multiplied by itself.
Question 6: State some application of the quadratic equation formula in real life?
Answer: In daily life, we use quadratic equation formula as for calculating areas, determining a product’s profit or formulating the speed of an object. In addition, quadratic equations refer to an equation that has at least one squared variable.
Question 7: What does a negative discriminant mean?
Answer: A positive discriminant denotes that the quadratic has two different real number solutions. A discriminant of zero denotes that the quadratic consists of a repeated real number solution. A negative discriminant denotes that neither of the solutions is real numbers.
Question 8: What is a negative quadratic?
Answer: A quadratic expression that always takes positive values is referred to as positive definite, while one that always takes negative values is referred to as negative definite. Moreover, quadratics of either type do not ever take the value 0, thus their discriminant is negative.
Question 9: Why is the discriminant important?
Answer: The quadratic equation discriminant is significant since it tells us the number and kind of solutions. This information is useful as it serves as a double check when we solve quadratic equations by any of the four techniques. The four techniques are factoring, completing the square, using square roots, and using the quadratic equation formula.
Question 10: What is the nature of roots of quadratic equation?
Answer: The nature of roots of quadratic equation simply means the category in which the roots are falling upon. The roots may be imaginary, real, unequal or equal. If the discriminant is negative, the roots will be imaginary.