Question

use the discriminant to answer the questions
12x - 4 = 4x² + 60
for the equation shown, choose the description of the solutions
☑️ one real, rational
❌ two real, rational
❌ two real, irrational
two complex
complete

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
done

Explanation:

First Problem: $12x - 4 - 4x^2 = 0$

Step1: Rewrite in standard form

Rearrange to $ax^2+bx+c=0$:
$$-4x^2 + 12x - 4 = 0$$
Multiply by $-1$: $4x^2 - 12x + 4 = 0$

Step2: Identify a, b, c

$a=4$, $b=-12$, $c=4$

Step3: Calculate discriminant

Use $\Delta = b^2 - 4ac$:
$$\Delta = (-12)^2 - 4(4)(4) = 144 - 64 = 80$$

Step4: Analyze discriminant

$\Delta=80>0$, and $\sqrt{80}=4\sqrt{5}$ (irrational). So two real, irrational solutions.

Step1: Simplify to standard form

Combine constants and rearrange:
$$y = -0.5x^2 + 6x + 2$$

Step2: Identify a, b, c

$a=-0.5$, $b=6$, $c=2$

Step3: Calculate discriminant

Use $\Delta = b^2 - 4ac$:
$$\Delta = 6^2 - 4(-0.5)(2) = 36 + 4 = 40$$

Step4: Analyze discriminant

$\Delta=40>0$, so the graph has two x-intercepts.

Answer:

Two real, irrational

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Second Problem: $y = 5 - 0.5x^2 + 6x - 3$