Answer
Verified
413.1k+ views
Hint:When we say there is no $x$ -intercept it means that it does not cross the $x$ -axis. In other words, we can say that if a line has no $x$ -intercept, then it never intersects $x$ -axis which means that it is parallel to $y$ -axis. So, we can conclude that this is a vertical line and its slope is undefined. Therefore, the possibility of two solutions is definitely ruled out.
Complete step by step solution:
Here, in this question students have to determine whether there are two, one or no real solutions for the graph of a quadratic function which does not have a $x$ -intercept.
Let us assume that we include a point of coincidence i.e., the vertex coincides with the $x$ -axis, then the plot doesn’t cross the $x$-axis nor does any point on the curve coincide with it. In such an assumption, there are no real solutions.
The $x$ -axis is composed of all those points for which $f\left( x \right)$ is equal to zero.
If the graph of $f\left( x \right)$ doesn’t have an $x$ -intercept then, it means that it has no real solutions or points for which $f\left( x \right) = 0$.
Hence, we conclude that if the graph of a quadratic function does not have an $x$ -intercept then, it has no real solutions or roots.
Note: Students can also check whether a quadratic equation has two, one or no real solutions by using a quadratic formula. In the quadratic formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, the expression$\sqrt {{b^2} - 4ac} $ is called as discriminant and is often denoted by $D$. If $D$ is positive or greater than zero, then the two roots of the equation are real. If $D$ is zero, then roots are real but if $D$ is negative or less than zero, then roots are not real.
Complete step by step solution:
Here, in this question students have to determine whether there are two, one or no real solutions for the graph of a quadratic function which does not have a $x$ -intercept.
Let us assume that we include a point of coincidence i.e., the vertex coincides with the $x$ -axis, then the plot doesn’t cross the $x$-axis nor does any point on the curve coincide with it. In such an assumption, there are no real solutions.
The $x$ -axis is composed of all those points for which $f\left( x \right)$ is equal to zero.
If the graph of $f\left( x \right)$ doesn’t have an $x$ -intercept then, it means that it has no real solutions or points for which $f\left( x \right) = 0$.
Hence, we conclude that if the graph of a quadratic function does not have an $x$ -intercept then, it has no real solutions or roots.
Note: Students can also check whether a quadratic equation has two, one or no real solutions by using a quadratic formula. In the quadratic formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, the expression$\sqrt {{b^2} - 4ac} $ is called as discriminant and is often denoted by $D$. If $D$ is positive or greater than zero, then the two roots of the equation are real. If $D$ is zero, then roots are real but if $D$ is negative or less than zero, then roots are not real.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE