To find the x-intercepts of the quadratic equation y = 2x² + 3x – 20, we need to set y to 0 and solve for x. This means we are looking for the points where the graph of the equation crosses the x-axis.
So, we start with the equation:
0 = 2x² + 3x - 20Next, we can either factor this quadratic or use the quadratic formula. Let’s first try factoring:
We are looking for two numbers that multiply to (2 * -20) = -40 and add up to 3. The numbers 8 and -5 work since:
- 8 * -5 = -40
- 8 + (-5) = 3
Now, we can rewrite the middle term (3x) using these two numbers:
0 = 2x² + 8x - 5x - 20Next, we can factor by grouping:
0 = (2x² + 8x) + (-5x - 20)Factoring out common terms in each group gives us:
0 = 2x(x + 4) - 5(x + 4)Now, we can factor out the common factor:
0 = (2x - 5)(x + 4)Setting each factor equal to zero gives us the x-intercepts:
2x - 5 = 0 or x + 4 = 0Solving these equations:
- 2x – 5 = 0:
x = 5/2 = 2.5 - x + 4 = 0:
x = -4
Thus, the x-intercepts of the equation y = 2x² + 3x – 20 are (2.5, 0) and (-4, 0).