Question

find the discriminant and use it to determine the number of x - intercepts of the graph of f. also, determine the number of real solutions to the equation f(x)=0. f(x)=x^2 + 3x + 6 discriminant: which means there are x - intercept(s) and there are real solution(s) to the equation f(x)=0. question help: ebook written example

Explanation:

Step1: Identify coefficients

For the quadratic function $f(x)=x^{2}+3x + 6$, where $a = 1$, $b=3$, $c = 6$.

Step2: Calculate discriminant

The discriminant formula for a quadratic equation $ax^{2}+bx + c=0$ is $\Delta=b^{2}-4ac$. Substitute $a = 1$, $b = 3$, $c = 6$ into the formula: $\Delta=(3)^{2}-4\times1\times6=9 - 24=- 15$.

Step3: Determine number of solutions and x - intercepts

If $\Delta>0$, there are two real - solutions and two $x$-intercepts. If $\Delta = 0$, there is one real - solution and one $x$-intercept. If $\Delta<0$, there are no real - solutions and no $x$-intercepts. Since $\Delta=-15<0$, there are 0 $x$-intercepts and 0 real solutions to the equation $f(x)=0$.

Answer:

Discriminant: $-15$
There are 0 $x$-intercept(s) and there are 0 real solution(s) to the equation $f(x)=0$.