Question

Find the number of real solutions of the equation $3x^2=6x-9$ by using discriminant.

Short answer

The equation has no real solutions.

Step-by-step solution

Step 1. From discriminant method of quadratic equation.

An equation of the form$a{x}^2+bx+c=0$ where$a\ne 0$ has discriminant$\left(D\right)=b^2-4ac$ in which three particular cases arises.

1. If $D>0$, the equation has two distinct real solutions.
2. If $D<0$, the equation has no real solution.
3. If $D=0$, the equation has exactly one real solution.

Step 2. Evaluate discriminant of given quadratic.

The discriminant of given quadratic equation $3x^2=6x-9$which can be written as $3x^2-6x+9=0$is evaluated as,

role="math" localid="1643823925771" $$\begin{gathered} D={\left(-6\right)}^2-4\left(3\right)\left(9\right) \\ =36-108 \\ =-72 \\ <0....\left(1\right) \end{gathered}$$

Step 3. DESCRIPTION OF STEPS.

Since$D<0$ from (1), therefore the equation no real solutions.