91Ó°ÊÓ

The discriminant is a key concept when solving quadratic equations. It tells us about the nature of the roots of the equation.
The formula for the discriminant is \[b^2 - 4ac \]. Here, \(b\) is the coefficient of the linear term, \(a\) is the coefficient of the quadratic term, and \(c\) is the constant term.
For our equation \(36 x^2 + 84 x + 49 = 0\), we have \(a = 36\), \(b = 84\), and \(c = 49\).
Let's calculate the discriminant:
\[Discriminant = 84^2 - 4(36)(49) \]Perform the calculations:
\[84^2 = 7056\] and \[4(36)(49) = 7056 \].
So,
\[Discriminant = 7056 - 7056 = 0\]
This zero value indicates that the quadratic equation has one unique real solution or a repeated root.

The quadratic formula is essential for solving quadratic equations. It is given by \[-\frac{b \text{±} \sqrt{b^2 - 4ac}}{2a} \].
This formula helps find the roots of any quadratic equation \(ax^2 + bx + c = 0\).
In our example, we use the coefficients we identified earlier: \(a = 36\), \(b = 84\), and \(c = 49\).
Since we already calculated the discriminant to be 0, we can plug these into the quadratic formula:
\[-\frac{84 \text{±} \sqrt{0}}{2(36)} \]With the discriminant being 0, the term under the square root becomes zero, simplifying our formula to:
\[-\frac{84}{72} \] Simplify this fraction further:
\[-\frac{84}{72} = -\frac{7}{6}\] So, the root of the equation is \(x = -\frac{7}{6}\).
It's a single repeated root because the discriminant is zero.

Repeated roots occur when the discriminant of the quadratic equation is zero.
A quadratic equation usually has two solutions for \(x\), but if the discriminant is zero, the equation has only one unique solution, referred to as a repeated root.
In our example with equation \(36 x^2 + 84 x + 49 = 0\), the discriminant turned out to be zero:
\[84^2 - 4(36)(49) = 7056 - 7056 = 0 \]This means the equation has one solution.
Using the quadratic formula, we found this solution to be:
\[-\frac{84}{72} = -\frac{7}{6}\] Here, \(x = -\frac{7}{6}\) is the repeated root.
Repeated root means the graph of the quadratic equation touches the x-axis at only one point, and this point is \(x = -\frac{7}{6}\).