Question

How many solutions does the equation $6{x^2} - 2x = - 9$ have?

Answer 1

Hint: The question belongs to the quadratic equation. To find the number of solutions of the given equation, first we will calculate the discriminant of the given quadratic equation. We will use the formula of discriminant. The formula of discriminant is given by the equation $D = {b^2} - 4ac$ where D is the discriminant of the quadratic equation. To calculate the discriminant of the given equation, we will compare the given quadratic equation with the standard equation of a quadratic. The standard equation of a quadratic equation is $f\left( x \right) = a{x^2} + bx + c$ , where $a$ Is the leading coefficient of the equation and $c$ is the absolute term of the quadratic equation. If the discriminant is equal to zero, then the quadratic equation will have only one solution. If the discriminant is greater than zero, then the quadratic equation will have two solutions. If the discriminant is less than zero, then the quadratic equation will have no real solutions.

Complete step by step solution:
Step: 1 the given quadratic equation is,
$6{x^2} - 2x = - 9$
Bring constant terms of the given equation to the right hand side of the equation.
$ \Rightarrow 6{x^2} - 2x + 9 = 0$
 Step: 2 Now compare the given quadratic equation with the general form of equation of a quadratic equation.
The general form of equation of quadratic is,
$ \Rightarrow f\left( x \right) = a{x^2} + bx + c$
Therefore,
$
   \Rightarrow a = 6 \\
   \Rightarrow b = - 2 \\
   \Rightarrow c = 9 \\
 $
The formula of discriminant is,
$ \Rightarrow D = {b^2} - 4ac$
Substitute the value in the given formula to find the discriminant of the given quadratic equation.
$ \Rightarrow D = {\left( { - 2} \right)^2} - 4 \times 6 \times 9$
Solve the equation to find the value of discriminant.
$
   \Rightarrow D = 4 - 216 \\
   \Rightarrow D = - 212 \\
 $
As we can see that the value of discriminant is less than zero, so the given quadratic equation will have no real solutions.


Therefore the given quadratic equation will have no real solutions.

Note:
Students are advised to remember the property discriminant. If the discriminant of a quadratic equation is less than zero, then it will have no real solutions. And if the discriminant of a quadratic equation is greater than zero, then the equation will have two real solutions. Use the discriminant formula to calculate the value of discriminant.