+1859 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.
+121 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:
\[m \in (-\infty, -4) \cup (12, \infty).\]
+121 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:
\[m \in (-\infty, -4) \cup (12, \infty).\]
