This is a tutorial on how to solve quadratic equations graphically and check the answers analytically. The quadratic equations explored are of the type:
\[ ax^2 + bx + c = 0 \]
The analytical solutions to a quadratic equation are given by the quadratic formula:
\[ x_1 = \frac{-b + \sqrt{\Delta}}{2a}, \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a} \]
where \(\Delta = b^2 - 4ac\) is called the discriminant. It gives information about the number and type of solutions:
The graphical solutions are obtained by graphing the function \(y = ax^2 + bx + c\). The x-intercepts, if they exist, provide approximate solutions.
Solve graphically and analytically:
\[2x^2 + 3x = 5\]Rewrite in standard form:
\[ 2x^2 + 3x - 5 = 0 \]
Graph the function:
\[ y = 2x^2 + 3x - 5 \]
Locate x-intercepts: \(x_1 \approx -2.5\), \(x_2 \approx 1\).
Discriminant:
\[ \Delta = b^2 - 4ac = 3^2 - 4(2)(-5) = 49 \]
Since \(\Delta > 0\), two real solutions exist:
\[ x_1 = \frac{-3 + \sqrt{49}}{2\cdot 2} = 1, \quad x_2 = \frac{-3 - \sqrt{49}}{2\cdot 2} = -2.5 \]
The graphical and analytical solutions match. Graphical solutions are approximate in general.
Solve graphically and analytically:
\[ x^2 + 4x + 4 = 0 \]Graph the function:
\[ y = x^2 + 4x + 4 \]
There is one x-intercept: \(x = -2\).
Discriminant:
\[ \Delta = b^2 - 4ac = 16 - 16 = 0 \]
Since \(\Delta = 0\), one solution exists:
\[ x = \frac{-b}{2a} = \frac{-4}{2 \cdot 1} = -2 \]
Solve graphically and analytically:
\[ -x^2 + 4x - 5 = 0 \]Graph the function:
\[ y = -x^2 + 4x - 5 \]
No x-intercepts; no real solutions, only complex ones.
Discriminant:
\[ \Delta = b^2 - 4ac = 4^2 - 4(-1)(-5) = -4 \]
Since \(\Delta < 0\), two complex conjugate solutions exist:
\[ x_1 = \frac{-4 + \sqrt{-4}}{2(-1)} = 2 - i, \quad x_2 = \frac{-4 - \sqrt{-4}}{2(-1)} = 2 + i \]
Graphical method cannot find imaginary solutions.
Solve graphically and analytically: