This tutorial explains how to find the points of intersection between a parabola and a line given by their equations.
Find the points of intersection of the parabola and the line given by:
\[ y = 2x^2 + 4x - 3 \] \[ 2y + x = 4 \]
Step 1: Solve the linear equation for \(y\):
\[ 2y + x = 4 \implies y = -\frac{1}{2}x + 2 \]Step 2: Substitute \(y = -\frac{1}{2}x + 2\) into the parabola equation:
\[ -\frac{1}{2}x + 2 = 2x^2 + 4x - 3 \]Step 3: Bring all terms to one side to form a quadratic equation:
\[ 2x^2 + \frac{9}{2}x - 5 = 0 \]Step 4: Solve the quadratic equation using the quadratic formula:
\[ x = \frac{-9 \pm \sqrt{241}}{8} \]Step 5: Find corresponding \(y\)-values by substituting \(x\) back into \(y = -\frac{1}{2}x + 2\):
\[ y = \frac{41 \pm \sqrt{241}}{16} \]Step 6: Points of intersection:
\[ \left( \frac{-9 - \sqrt{241}}{8}, \frac{41 + \sqrt{241}}{16} \right), \quad \left( \frac{-9 + \sqrt{241}}{8}, \frac{41 - \sqrt{241}}{16} \right) \]Approximate coordinates:
\[ (-3.06, 3.53) \quad \text{and} \quad (0.82, 1.59) \]Graph of the parabola, line, and intersection points: