Question:
At which of the following points, the quadratic polynomial \(p(x) = - 3x + 18x^2 - 1\) intersects the positive x-axis ?
At which of the following points, the quadratic polynomial \(p(x) = - 3x + 18x^2 - 1\) intersects the positive x-axis ?
Show Hint
Always read the constraint "positive x-axis" carefully.
Both potential answers \(\left(-\frac{1}{6}, 0\right)\) and \(\left(\frac{1}{3}, 0\right)\) are present in the options, but only one is positive!
Both potential answers \(\left(-\frac{1}{6}, 0\right)\) and \(\left(\frac{1}{3}, 0\right)\) are present in the options, but only one is positive!
Updated On: Jul 9, 2026
- \(\left(\frac{1}{6}, 0\right)\)
- \(\left(-\frac{1}{3}, 0\right)\)
- \(\left(-\frac{1}{6}, 0\right)\)
- \(\left(\frac{1}{3}, 0\right)\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
The question asks for the point where the quadratic polynomial \(p(x) = 18x^2 - 3x - 1\) intersects the positive x-axis.
Any point on the x-axis has a y-coordinate (or value of the polynomial) equal to zero.
Therefore, we must find the roots of the equation \(p(x) = 0\) and select the positive root.
Step 2: Key Formula or Approach:
For a quadratic polynomial \(ax^2 + bx + c\), the intersections with the x-axis are found by solving:
\[ ax^2 + bx + c = 0 \]
We can solve this quadratic equation using factorization (splitting the middle term).
Step 3: Detailed Explanation:
• Write down the quadratic equation:
\[ 18x^2 - 3x - 1 = 0 \]
• Identify the product and sum needed for splitting the middle term:
Product = \(18 \times (-1) = -18\)
Sum = \(-3\)
• The factors of \(-18\) that add up to \(-3\) are \(-6\) and \(+3\).
• Rewrite the middle term using these factors:
\[ 18x^2 - 6x + 3x - 1 = 0 \]
• Factor by grouping:
\[ 6x(3x - 1) + 1(3x - 1) = 0 \]
\[ (6x + 1)(3x - 1) = 0 \]
• Set each factor to zero to find the roots:
\[ 6x + 1 = 0 \implies x = -\frac{1}{6} \]
\[ 3x - 1 = 0 \implies x = \frac{1}{3} \]
• Since we are looking for the intersection with the positive x-axis, we select the positive value:
\[ x = \frac{1}{3} \]
• Thus, the coordinates of the point of intersection are:
\[ \left(\frac{1}{3}, 0\right) \]
Step 4: Final Answer:
The polynomial intersects the positive x-axis at the point \(\left(\frac{1}{3}, 0\right)\).
The question asks for the point where the quadratic polynomial \(p(x) = 18x^2 - 3x - 1\) intersects the positive x-axis.
Any point on the x-axis has a y-coordinate (or value of the polynomial) equal to zero.
Therefore, we must find the roots of the equation \(p(x) = 0\) and select the positive root.
Step 2: Key Formula or Approach:
For a quadratic polynomial \(ax^2 + bx + c\), the intersections with the x-axis are found by solving:
\[ ax^2 + bx + c = 0 \]
We can solve this quadratic equation using factorization (splitting the middle term).
Step 3: Detailed Explanation:
• Write down the quadratic equation:
\[ 18x^2 - 3x - 1 = 0 \]
• Identify the product and sum needed for splitting the middle term:
Product = \(18 \times (-1) = -18\)
Sum = \(-3\)
• The factors of \(-18\) that add up to \(-3\) are \(-6\) and \(+3\).
• Rewrite the middle term using these factors:
\[ 18x^2 - 6x + 3x - 1 = 0 \]
• Factor by grouping:
\[ 6x(3x - 1) + 1(3x - 1) = 0 \]
\[ (6x + 1)(3x - 1) = 0 \]
• Set each factor to zero to find the roots:
\[ 6x + 1 = 0 \implies x = -\frac{1}{6} \]
\[ 3x - 1 = 0 \implies x = \frac{1}{3} \]
• Since we are looking for the intersection with the positive x-axis, we select the positive value:
\[ x = \frac{1}{3} \]
• Thus, the coordinates of the point of intersection are:
\[ \left(\frac{1}{3}, 0\right) \]
Step 4: Final Answer:
The polynomial intersects the positive x-axis at the point \(\left(\frac{1}{3}, 0\right)\).
Was this answer helpful?
0
0
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 \(n\).
View More Questions
Top CBSE X Geometrical Meaning of the Zeroes of a Polynomial Questions
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) \(x^2 – 2x – 8\) (ii) \(4s^2 – 4s + 1\) (iii) \(6x^2 – 3 – 7x\) (iv) \(4u^2 + 8u\) (v) \( t^2 – 15\) (vi) \(3x^2 – x – 4\)
- The graphs of \(y = p(x)\) are given in Fig. 2.10 below, for some polynomials \(p(x)\) . Find the number of zeroes of \(p(x)\), in each case
(I) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) - The graph of a polynomial \(p(x)\) is shown here. The number of zeroes of the polynomial \(p(x)\) is
- The graph of $y = f(x)$ is given. The number of zeroes of $f(x)$ is :
- The graph of a quadratic polynomial f(x) passes through (5,0), (0, -1) and (-2, 0). The two factors of the polynomial are
View More Questions
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
View More Questions