Question

y - 5 = 0.5x² + 6x - 3
how many x-intercepts does the graph of this quadratic have?
two x-intercepts
one x-intercept
no x-intercepts

Explanation:

Step1: Rearrange to standard form

First, isolate $y$ and simplify the quadratic:
$y = 0.5x^2 + 6x - 3 + 5$
$y = 0.5x^2 + 6x + 2$
For x-intercepts, set $y=0$:
$0.5x^2 + 6x + 2 = 0$
Multiply all terms by 2 to eliminate decimals:
$x^2 + 12x + 4 = 0$

Step2: Calculate discriminant

For quadratic $ax^2+bx+c=0$, discriminant $D = b^2-4ac$.
Here, $a=1$, $b=12$, $c=4$:
$D = 12^2 - 4(1)(4)$
$D = 144 - 16 = 128$

Step3: Analyze discriminant

Since $D=128>0$, the quadratic has two distinct real roots, meaning two x-intercepts.

Answer:

Two x-intercepts