Questions: Compute the discriminant of (6 x^2+6=-6 x). The discriminant is: (square) How many real solutions does the equation, (6 x^2+6=-6 x), have? No real solution One real solution Two real solutions None of the above

Compute the discriminant of (6 x^2+6=-6 x).
The discriminant is: (square)
How many real solutions does the equation, (6 x^2+6=-6 x), have?
No real solution
One real solution
Two real solutions
None of the above

Transcript text: Compute the discriminant of $6 x^{2}+6=-6 x$. The discriminant is: $\square$ How many real solutions does the equation, $6 x^{2}+6=-6 x$, have? No real solution One real solution Two real solutions None of the above

Solution

Solution Steps

To solve the given quadratic equation $6x^2 + 6 = -6x$, we first need to rewrite it in the standard form $ax^2 + bx + c = 0$. Then, we can compute the discriminant using the formula $\Delta = b^2 - 4ac$. The discriminant will help us determine the number of real solutions: if $\Delta > 0$, there are two real solutions; if $\Delta = 0$, there is one real solution; and if $\Delta < 0$, there are no real solutions.

Solution Approach

Rewrite the equation in standard form.
Identify the coefficients $a$, $b$, and $c$.
Compute the discriminant $\Delta = b^2 - 4ac$.
Determine the number of real solutions based on the value of $\Delta$.

Step 1: Rewrite the Equation

The given equation is $6x^2 + 6 = -6x$. We can rewrite it in standard form as: \[ 6x^2 + 6x + 6 = 0 \]

Step 2: Identify Coefficients

From the standard form $ax^2 + bx + c = 0$, we identify the coefficients: \[ a = 6, \quad b = 6, \quad c = 6 \]

Step 3: Compute the Discriminant

The discriminant $\Delta$ is calculated using the formula: \[ \Delta = b^2 - 4ac \] Substituting the values of $a$, $b$, and $c$: \[ \Delta = 6^2 - 4 \cdot 6 \cdot 6 = 36 - 144 = -108 \]

Step 4: Determine the Number of Real Solutions

Since the discriminant $\Delta = -108$ is less than zero, this indicates that there are no real solutions to the equation.