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Let $m$ be a real number.  If the quadratic equation $x^2+mx+4 = 4x - 12$ has two distinct real roots, then what are the possible values of $m$?  Express your answer in interval notation.

Aug 15, 2023

#2
+121
+1

Given the quadratic equation $$x^2 + mx + 4 = 4x - 12$$, we can rearrange it into standard quadratic form:

$x^2 + (m - 4)x + 16 = 0.$

For this quadratic equation to have two distinct real roots, the discriminant ($$b^2 - 4ac$$) must be greater than zero, where $$a = 1$$ (coefficient of $$x^2$$), $$b = m - 4$$ (coefficient of $$x$$), and $$c = 16$$ (constant term).

The discriminant condition is:

$(m - 4)^2 - 4 \cdot 1 \cdot 16 > 0.$

Simplify the inequality:

$m^2 - 8m + 16 - 64 > 0.$
$m^2 - 8m - 48 > 0.$

Now, we can factor the quadratic:

$(m - 12)(m + 4) > 0.$

To determine the intervals where this inequality is satisfied, we can analyze the sign of the expression $$m - 12$$ and $$m + 4$$:

1. When $$m < -4$$, both factors $$m - 12$$ and $$m + 4$$ are negative. So, the expression is positive: $$(m - 12)(m + 4) > 0$$.

2. When $$-4 < m < 12$$, the factor $$m - 12$$ is negative, but $$m + 4$$ is positive. So, the expression is negative: $$(m - 12)(m + 4) < 0$$.

3. When $$m > 12$$, both factors $$m - 12$$ and $$m + 4$$ are positive. So, the expression is positive: $$(m - 12)(m + 4) > 0$$.

So, the values of $$m$$ that satisfy the inequality are $$m < -4$$ or $$m > 12$$. This can be expressed in interval notation as:

$m \in (-\infty, -4) \cup (12, \infty).$

Aug 16, 2023

#1
0

8+8=16

Aug 15, 2023
#2
+121
+1

Given the quadratic equation $$x^2 + mx + 4 = 4x - 12$$, we can rearrange it into standard quadratic form:

$x^2 + (m - 4)x + 16 = 0.$

For this quadratic equation to have two distinct real roots, the discriminant ($$b^2 - 4ac$$) must be greater than zero, where $$a = 1$$ (coefficient of $$x^2$$), $$b = m - 4$$ (coefficient of $$x$$), and $$c = 16$$ (constant term).

The discriminant condition is:

$(m - 4)^2 - 4 \cdot 1 \cdot 16 > 0.$

Simplify the inequality:

$m^2 - 8m + 16 - 64 > 0.$
$m^2 - 8m - 48 > 0.$

Now, we can factor the quadratic:

$(m - 12)(m + 4) > 0.$

To determine the intervals where this inequality is satisfied, we can analyze the sign of the expression $$m - 12$$ and $$m + 4$$:

1. When $$m < -4$$, both factors $$m - 12$$ and $$m + 4$$ are negative. So, the expression is positive: $$(m - 12)(m + 4) > 0$$.

2. When $$-4 < m < 12$$, the factor $$m - 12$$ is negative, but $$m + 4$$ is positive. So, the expression is negative: $$(m - 12)(m + 4) < 0$$.

3. When $$m > 12$$, both factors $$m - 12$$ and $$m + 4$$ are positive. So, the expression is positive: $$(m - 12)(m + 4) > 0$$.

So, the values of $$m$$ that satisfy the inequality are $$m < -4$$ or $$m > 12$$. This can be expressed in interval notation as:

$m \in (-\infty, -4) \cup (12, \infty).$

SpectraSynth Aug 16, 2023