Check whether the following are quadratic equations :
(i) \((x + 1)^2 = 2(x – 3)\) (ii) \(x^2 – 2x = (–2) (3 – x)\)
(iii) \((x – 2)(x + 1) = (x – 1)(x + 3)\) (iv) \((x – 3)(2x +1) = x(x + 5)\)
(v) \((2x – 1)(x – 3) = (x + 5)(x – 1)\) (vi) \(x^2 + 3x + 1 = (x – 2)^2\)
(vii) \((x + 2)^3 = 2x (x^2 – 1)\) (viii) \(x^3 – 4x^2 – x + 1 = (x – 2)^3\)
(i) \((x + 1)^2 = 2(x – 3)\) (ii) \(x^2 – 2x = (–2) (3 – x)\)
(iii) \((x – 2)(x + 1) = (x – 1)(x + 3)\) (iv) \((x – 3)(2x +1) = x(x + 5)\)
(v) \((2x – 1)(x – 3) = (x + 5)(x – 1)\) (vi) \(x^2 + 3x + 1 = (x – 2)^2\)
(vii) \((x + 2)^3 = 2x (x^2 – 1)\) (viii) \(x^3 – 4x^2 – x + 1 = (x – 2)^3\)
Solution and Explanation
(i) \((x + 1)^2 = 2(x – 3) ⇒ x^2 + 2x + 1 = 2x -6 ⇒ x^2 +7 =0\)
It is of the form \(ax^2 + bx +c =0.\)
Hence, the given equation is a quadratic equation.
(ii) \(x^2 – 2x = (–2) (3 – x) ⇒ x^2 -2x = -6 + 2x ⇒ x^2 -4x +6\)
It is of the form \(ax^2 + bx +c =0.\)
Hence, the given equation is a quadratic equation.
(iii) \((x – 2)(x + 1) = (x – 1)(x + 3) ⇒ x^2 -x-2 = x^2 +2x -3 ⇒ 3x-1\)
It is not of the form \(ax^2 + bx +c =0.\)
Hence, the given equation is not a quadratic equation.
(iv) \((x – 3)(2x +1) = x(x + 5) ⇒ 2x^2 -5x -3 = x^2 +5x ⇒ x^2 -10x -3\)
It is of the form \(ax^2 + bx +c =0.\)
Hence, the given equation is a quadratic equation.
(v) \((2x – 1)(x – 3) = (x + 5)(x – 1) ⇒ 2x^2 -7x +3 = x^2 +4x -5 ⇒ x^2-11x +8 =0\)
It is of the form \(ax^2 + bx +c =0.\)
Hence, the given equation is a quadratic equation.
(vi) \(x^2 + 3x + 1 = (x – 2)^2 ⇒ x^2 +3x +1 = x^2 +4 -4x ⇒7x -3 =0\)
It is not of the form \(ax^2 + bx +c =0.\)
Hence, the given equation is not a quadratic equation.
(vii) \((x + 2)^3 = 2x (x^2 – 1) ⇒x^3 +8 +6x^2 +12x ⇒ 2x^3 -2x ⇒ x^2 -14x -6x^2 -8=0\)
It is of the form \(ax^2 + bx +c =0.\)
Hence, the given equation is a quadratic equation.
(viii) \(x^3 – 4x^2 – x + 1 = (x – 2)^3 ⇒ x^3 -4x^2 -x +1 = x^3 -8-6x^2 +12x ⇒ 2x^2 -13x +9\)
It is of the form \(ax^2 + bx +c =0.\)
Hence, the given equation is a quadratic equation.
Top CBSE X Mathematics Questions
- Express each number as a product of its prime factors:
- \(140\)
- \(156\)
- \(3825\)
- \(5005\)
- \(7429\)
- Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
- \(26\) and \(91\)
- \(510\) \(\)and \(92\)
- \(336\) and \(54\)
- Find the LCM and HCF of the following integers by applying the prime factorisation method.
- \(12, 15\) and
- \(17, 23\) and
- \(8, 9\) and \(25\)
- Given that HCF \((306, 657) = 9\) , find LCM \((306, 657)\).
- Check whether \(6n\) can end with the digit \(0\) for any natural number \(n\).
Top CBSE X Quadratic Equations Questions
Represent the following situations in the form of quadratic equations.
(i) The area of a rectangular plot is 528 \(m^2\). The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
- If the roots of quadratic equation \(4x^2 - 5x + k = 0\) are real and equal, then value of \(k\) is:
- The roots of the quadratic equation \(x^2 + x – p (p + 1) = 0 \)are :
- If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = 6x^2 + 11x - 10$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.
- If the roots of quadratic equation \(4x^2 - 5x + k = 0\) are real and equal, then value of \(k\) is:
Top CBSE X Questions
- Read the following source carefully and answer the questions that follow:
Print Comes to India
From 1780, James Augustus Hickey began to edit the Bengal Gazette, a weekly maga zine that described itself as ‘a commercial paper open to all, but influenced by none’. So it was private English enterprise, proud of its independence from colonial influence, that began English printing in India. Hickey published a lot of advertisements, including those that related to the import and sale of slaves. But he also published a lot of gossip about the Company’s senior officials in India. Enraged by this, Governor-General War ren Hastings persecuted Hickey, and encouraged the publication of officially sanctioned newspapers that could counter the flow of information that damaged the image of the colonial government. By the close of the eighteenth century, a number of newspapers and journals appeared in print. There were Indians, too, who began to publish Indian news papers. The first to appear was the weekly Bengal Gazette, brought out by Gangadhar Bhattacharya, who was close to Raja Rammohan Roy. - Read the following source carefully and answer the questions that follow:
Loans from Cooperatives
Besides banks, the other major source of cheap credit in rural areas are the cooperative societies (or cooperatives). Members of a cooperative pool their resources for coopera tion in certain areas. There are several types of cooperatives possible such as farmers cooperatives, weavers cooperatives, industrial workers cooperatives, etc. Krishak Coop erative functions in a village not very far away from Sonpur. It has 2300 farmers as members. It accepts deposits from its members. With these deposits as collateral, the Cooperative has obtained a large loan from the bank. These funds are used to provide loans to members. Once these loans are repaid, another round of lending can take place. Krishak Cooperative provides loans for the purchase of agricultural implements, loans for cultivation and agricultural trade, fishery loans, loans for construction of houses and for a variety of other expenses. - More than three million carbon compounds have been discovered in the field of chemistry. The diversity of these compounds is due to the capacity of carbon atoms for bonding with one another as well as with other atoms. Most of the carbon compounds are poor conductors of electricity and have low melting and boiling points.
- Human digestive system is a tube running from mouth to anus. Its main function is to breakdown complex molecules present in the food which cannot be absorbed as such into smaller molecules. These molecules are absorbed across the walls of the tube and the absorbed food reaches each and every cell of the body where it is utilised for obtaining energy.
- A student has three convex lenses A, B, and C of different focal lengths. He wants to observe the images formed by these lenses on a screen by placing a candle flame at different distances as given in the following table:
Case No. Lens Focal Length Object Distance 1 \(A\) 50 cm 25 cm 2 B 20 cm 60 cm 3 C 15 cm 30 cm
Concepts Used:
Quadratic Equations
A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)
Two important points to keep in mind are:
- A polynomial equation has at least one root.
- A polynomial equation of degree ‘n’ has ‘n’ roots.
Read More: Nature of Roots of Quadratic Equation
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root